Integral de $$$84 i n t \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)}$$$ con respecto a $$$x$$$

Halla $$$\int 84 i n t \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)}\, dx$$$.

Aplica la fórmula de reducción de potencia $$$\cos^{3}{\left(\alpha \right)} = \frac{3 \cos{\left(\alpha \right)}}{4} + \frac{\cos{\left(3 \alpha \right)}}{4}$$$ con $$$\alpha=x$$$:

$${\color{red}{\int{84 i n t \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)} d x}}} = {\color{red}{\int{21 i n t \left(3 \cos{\left(x \right)} + \cos{\left(3 x \right)}\right) \sin^{3}{\left(x \right)} d x}}}$$

Aplica la fórmula de reducción de potencia $$$\sin^{3}{\left(\alpha \right)} = \frac{3 \sin{\left(\alpha \right)}}{4} - \frac{\sin{\left(3 \alpha \right)}}{4}$$$ con $$$\alpha=x$$$:

$${\color{red}{\int{21 i n t \left(3 \cos{\left(x \right)} + \cos{\left(3 x \right)}\right) \sin^{3}{\left(x \right)} d x}}} = {\color{red}{\int{\frac{21 i n t \left(3 \sin{\left(x \right)} - \sin{\left(3 x \right)}\right) \left(3 \cos{\left(x \right)} + \cos{\left(3 x \right)}\right)}{4} d x}}}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\frac{1}{16}$$$ y $$$f{\left(x \right)} = 84 i n t \left(3 \sin{\left(x \right)} - \sin{\left(3 x \right)}\right) \left(3 \cos{\left(x \right)} + \cos{\left(3 x \right)}\right)$$$:

$${\color{red}{\int{\frac{21 i n t \left(3 \sin{\left(x \right)} - \sin{\left(3 x \right)}\right) \left(3 \cos{\left(x \right)} + \cos{\left(3 x \right)}\right)}{4} d x}}} = {\color{red}{\left(\frac{\int{84 i n t \left(3 \sin{\left(x \right)} - \sin{\left(3 x \right)}\right) \left(3 \cos{\left(x \right)} + \cos{\left(3 x \right)}\right) d x}}{16}\right)}}$$

Expand the expression:

$$\frac{{\color{red}{\int{84 i n t \left(3 \sin{\left(x \right)} - \sin{\left(3 x \right)}\right) \left(3 \cos{\left(x \right)} + \cos{\left(3 x \right)}\right) d x}}}}{16} = \frac{{\color{red}{\int{\left(756 i n t \sin{\left(x \right)} \cos{\left(x \right)} + 252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} - 252 i n t \sin{\left(3 x \right)} \cos{\left(x \right)} - 84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)}\right)d x}}}}{16}$$

Integra término a término:

$$\frac{{\color{red}{\int{\left(756 i n t \sin{\left(x \right)} \cos{\left(x \right)} + 252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} - 252 i n t \sin{\left(3 x \right)} \cos{\left(x \right)} - 84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)}\right)d x}}}}{16} = \frac{{\color{red}{\left(\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x} + \int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x} - \int{252 i n t \sin{\left(3 x \right)} \cos{\left(x \right)} d x} - \int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}\right)}}}{16}$$

Reescribe $$$\sin\left(3 x \right)\cos\left(x \right)$$$ utilizando la fórmula $$$\sin\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \sin\left(\alpha-\beta \right)+\frac{1}{2} \sin\left(\alpha+\beta \right)$$$ con $$$\alpha=3 x$$$ y $$$\beta=x$$$:

$$\frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\int{252 i n t \sin{\left(3 x \right)} \cos{\left(x \right)} d x}}}}{16} = \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\int{252 i n t \left(\frac{\sin{\left(2 x \right)}}{2} + \frac{\sin{\left(4 x \right)}}{2}\right) d x}}}}{16}$$

Desarrolla la expresión:

$$\frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\int{252 i n t \left(\frac{\sin{\left(2 x \right)}}{2} + \frac{\sin{\left(4 x \right)}}{2}\right) d x}}}}{16} = \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\int{\left(126 i n t \sin{\left(2 x \right)} + 126 i n t \sin{\left(4 x \right)}\right)d x}}}}{16}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(x \right)} = 252 i n t \sin{\left(2 x \right)} + 252 i n t \sin{\left(4 x \right)}$$$:

$$\frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\int{\left(126 i n t \sin{\left(2 x \right)} + 126 i n t \sin{\left(4 x \right)}\right)d x}}}}{16} = \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\left(\frac{\int{\left(252 i n t \sin{\left(2 x \right)} + 252 i n t \sin{\left(4 x \right)}\right)d x}}{2}\right)}}}{16}$$

Integra término a término:

$$\frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\int{\left(252 i n t \sin{\left(2 x \right)} + 252 i n t \sin{\left(4 x \right)}\right)d x}}}}{32} = \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\left(\int{252 i n t \sin{\left(2 x \right)} d x} + \int{252 i n t \sin{\left(4 x \right)} d x}\right)}}}{32}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=252 i n t$$$ y $$$f{\left(x \right)} = \sin{\left(2 x \right)}$$$:

$$- \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\int{252 i n t \sin{\left(2 x \right)} d x}}}}{32} = - \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\left(252 i n t \int{\sin{\left(2 x \right)} d x}\right)}}}{32}$$

Sea $$$u=2 x$$$.

Entonces $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{2}$$$.

Por lo tanto,

$$- \frac{63 i n t {\color{red}{\int{\sin{\left(2 x \right)} d x}}}}{8} - \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} = - \frac{63 i n t {\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}}}{8} - \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$- \frac{63 i n t {\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}}}{8} - \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} = - \frac{63 i n t {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{2}\right)}}}{8} - \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16}$$

La integral del seno es $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$$- \frac{63 i n t {\color{red}{\int{\sin{\left(u \right)} d u}}}}{16} - \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} = - \frac{63 i n t {\color{red}{\left(- \cos{\left(u \right)}\right)}}}{16} - \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16}$$

Recordemos que $$$u=2 x$$$:

$$\frac{63 i n t \cos{\left({\color{red}{u}} \right)}}{16} - \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} = \frac{63 i n t \cos{\left({\color{red}{\left(2 x\right)}} \right)}}{16} - \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=252 i n t$$$ y $$$f{\left(x \right)} = \sin{\left(4 x \right)}$$$:

$$\frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\int{252 i n t \sin{\left(4 x \right)} d x}}}}{32} = \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\left(252 i n t \int{\sin{\left(4 x \right)} d x}\right)}}}{32}$$

Sea $$$u=4 x$$$.

Entonces $$$du=\left(4 x\right)^{\prime }dx = 4 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{4}$$$.

La integral se convierte en

$$\frac{63 i n t \cos{\left(2 x \right)}}{16} - \frac{63 i n t {\color{red}{\int{\sin{\left(4 x \right)} d x}}}}{8} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} = \frac{63 i n t \cos{\left(2 x \right)}}{16} - \frac{63 i n t {\color{red}{\int{\frac{\sin{\left(u \right)}}{4} d u}}}}{8} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{4}$$$ y $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$\frac{63 i n t \cos{\left(2 x \right)}}{16} - \frac{63 i n t {\color{red}{\int{\frac{\sin{\left(u \right)}}{4} d u}}}}{8} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} = \frac{63 i n t \cos{\left(2 x \right)}}{16} - \frac{63 i n t {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{4}\right)}}}{8} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16}$$

La integral del seno es $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$$\frac{63 i n t \cos{\left(2 x \right)}}{16} - \frac{63 i n t {\color{red}{\int{\sin{\left(u \right)} d u}}}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} = \frac{63 i n t \cos{\left(2 x \right)}}{16} - \frac{63 i n t {\color{red}{\left(- \cos{\left(u \right)}\right)}}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16}$$

Recordemos que $$$u=4 x$$$:

$$\frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left({\color{red}{u}} \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16} = \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left({\color{red}{\left(4 x\right)}} \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}{16}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=84 i n t$$$ y $$$f{\left(x \right)} = \sin{\left(3 x \right)} \cos{\left(3 x \right)}$$$:

$$\frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}}}{16} = \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} - \frac{{\color{red}{\left(84 i n t \int{\sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}\right)}}}{16}$$

Sea $$$u=\sin{\left(3 x \right)}$$$.

Entonces $$$du=\left(\sin{\left(3 x \right)}\right)^{\prime }dx = 3 \cos{\left(3 x \right)} dx$$$ (los pasos pueden verse »), y obtenemos que $$$\cos{\left(3 x \right)} dx = \frac{du}{3}$$$.

La integral puede reescribirse como

$$\frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} - \frac{21 i n t {\color{red}{\int{\sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}}}{4} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} = \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} - \frac{21 i n t {\color{red}{\int{\frac{u}{3} d u}}}}{4} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{3}$$$ y $$$f{\left(u \right)} = u$$$:

$$\frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} - \frac{21 i n t {\color{red}{\int{\frac{u}{3} d u}}}}{4} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} = \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} - \frac{21 i n t {\color{red}{\left(\frac{\int{u d u}}{3}\right)}}}{4} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16}$$

Aplica la regla de la potencia $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=1$$$:

$$\frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} - \frac{7 i n t {\color{red}{\int{u d u}}}}{4} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16}=\frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} - \frac{7 i n t {\color{red}{\frac{u^{1 + 1}}{1 + 1}}}}{4} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16}=\frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} - \frac{7 i n t {\color{red}{\left(\frac{u^{2}}{2}\right)}}}{4} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16}$$

Recordemos que $$$u=\sin{\left(3 x \right)}$$$:

$$\frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} - \frac{7 i n t {\color{red}{u}}^{2}}{8} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16} = \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} - \frac{7 i n t {\color{red}{\sin{\left(3 x \right)}}}^{2}}{8} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}{16}$$

Reescribe $$$\sin\left(x \right)\cos\left(3 x \right)$$$ utilizando la fórmula $$$\sin\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \sin\left(\alpha-\beta \right)+\frac{1}{2} \sin\left(\alpha+\beta \right)$$$ con $$$\alpha=x$$$ y $$$\beta=3 x$$$:

$$- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{{\color{red}{\int{252 i n t \sin{\left(x \right)} \cos{\left(3 x \right)} d x}}}}{16} = - \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{{\color{red}{\int{252 i n t \left(- \frac{\sin{\left(2 x \right)}}{2} + \frac{\sin{\left(4 x \right)}}{2}\right) d x}}}}{16}$$

Desarrolla la expresión:

$$- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{{\color{red}{\int{252 i n t \left(- \frac{\sin{\left(2 x \right)}}{2} + \frac{\sin{\left(4 x \right)}}{2}\right) d x}}}}{16} = - \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{{\color{red}{\int{\left(- 126 i n t \sin{\left(2 x \right)} + 126 i n t \sin{\left(4 x \right)}\right)d x}}}}{16}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(x \right)} = - 252 i n t \sin{\left(2 x \right)} + 252 i n t \sin{\left(4 x \right)}$$$:

$$- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{{\color{red}{\int{\left(- 126 i n t \sin{\left(2 x \right)} + 126 i n t \sin{\left(4 x \right)}\right)d x}}}}{16} = - \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{{\color{red}{\left(\frac{\int{\left(- 252 i n t \sin{\left(2 x \right)} + 252 i n t \sin{\left(4 x \right)}\right)d x}}{2}\right)}}}{16}$$

Integra término a término:

$$- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{{\color{red}{\int{\left(- 252 i n t \sin{\left(2 x \right)} + 252 i n t \sin{\left(4 x \right)}\right)d x}}}}{32} = - \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{{\color{red}{\left(- \int{252 i n t \sin{\left(2 x \right)} d x} + \int{252 i n t \sin{\left(4 x \right)} d x}\right)}}}{32}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=252 i n t$$$ y $$$f{\left(x \right)} = \sin{\left(2 x \right)}$$$:

$$- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} - \frac{{\color{red}{\int{252 i n t \sin{\left(2 x \right)} d x}}}}{32} = - \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} - \frac{{\color{red}{\left(252 i n t \int{\sin{\left(2 x \right)} d x}\right)}}}{32}$$

La integral $$$\int{\sin{\left(2 x \right)} d x}$$$ ya ha sido calculada:

$$\int{\sin{\left(2 x \right)} d x} = - \frac{\cos{\left(2 x \right)}}{2}$$

Por lo tanto,

$$- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} - \frac{63 i n t {\color{red}{\int{\sin{\left(2 x \right)} d x}}}}{8} + \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} = - \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{16} + \frac{63 i n t \cos{\left(4 x \right)}}{32} - \frac{63 i n t {\color{red}{\left(- \frac{\cos{\left(2 x \right)}}{2}\right)}}}{8} + \frac{\int{252 i n t \sin{\left(4 x \right)} d x}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=252 i n t$$$ y $$$f{\left(x \right)} = \sin{\left(4 x \right)}$$$:

$$- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{{\color{red}{\int{252 i n t \sin{\left(4 x \right)} d x}}}}{32} = - \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} + \frac{{\color{red}{\left(252 i n t \int{\sin{\left(4 x \right)} d x}\right)}}}{32}$$

La integral $$$\int{\sin{\left(4 x \right)} d x}$$$ ya ha sido calculada:

$$\int{\sin{\left(4 x \right)} d x} = - \frac{\cos{\left(4 x \right)}}{4}$$

Por lo tanto,

$$- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{63 i n t {\color{red}{\int{\sin{\left(4 x \right)} d x}}}}{8} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16} = - \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8} + \frac{63 i n t \cos{\left(4 x \right)}}{32} + \frac{63 i n t {\color{red}{\left(- \frac{\cos{\left(4 x \right)}}{4}\right)}}}{8} + \frac{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}{16}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=756 i n t$$$ y $$$f{\left(x \right)} = \sin{\left(x \right)} \cos{\left(x \right)}$$$:

$$- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8} + \frac{{\color{red}{\int{756 i n t \sin{\left(x \right)} \cos{\left(x \right)} d x}}}}{16} = - \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8} + \frac{{\color{red}{\left(756 i n t \int{\sin{\left(x \right)} \cos{\left(x \right)} d x}\right)}}}{16}$$

Sea $$$w=\sin{\left(x \right)}$$$.

Entonces $$$dw=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$$$ (los pasos pueden verse »), y obtenemos que $$$\cos{\left(x \right)} dx = dw$$$.

Entonces,

$$- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8} + \frac{189 i n t {\color{red}{\int{\sin{\left(x \right)} \cos{\left(x \right)} d x}}}}{4} = - \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8} + \frac{189 i n t {\color{red}{\int{w d w}}}}{4}$$

Aplica la regla de la potencia $$$\int w^{n}\, dw = \frac{w^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=1$$$:

$$- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8} + \frac{189 i n t {\color{red}{\int{w d w}}}}{4}=- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8} + \frac{189 i n t {\color{red}{\frac{w^{1 + 1}}{1 + 1}}}}{4}=- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8} + \frac{189 i n t {\color{red}{\left(\frac{w^{2}}{2}\right)}}}{4}$$

Recordemos que $$$w=\sin{\left(x \right)}$$$:

$$- \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8} + \frac{189 i n t {\color{red}{w}}^{2}}{8} = - \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8} + \frac{189 i n t {\color{red}{\sin{\left(x \right)}}}^{2}}{8}$$

Por lo tanto,

$$\int{84 i n t \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)} d x} = \frac{189 i n t \sin^{2}{\left(x \right)}}{8} - \frac{7 i n t \sin^{2}{\left(3 x \right)}}{8} + \frac{63 i n t \cos{\left(2 x \right)}}{8}$$

Simplificar:

$$\int{84 i n t \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)} d x} = \frac{7 i n t \left(- 9 \cos^{2}{\left(x \right)} + \cos^{2}{\left(3 x \right)} + 17\right)}{8}$$

Añade la constante de integración:

$$\int{84 i n t \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)} d x} = \frac{7 i n t \left(- 9 \cos^{2}{\left(x \right)} + \cos^{2}{\left(3 x \right)} + 17\right)}{8}+C$$

$$$\int 84 i n t \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)}\, dx = \frac{7 i n t \left(- 9 \cos^{2}{\left(x \right)} + \cos^{2}{\left(3 x \right)} + 17\right)}{8} + C$$$A