Integral de $$$23 \cos^{3}{\left(35 x \right)}$$$

Halla $$$\int 23 \cos^{3}{\left(35 x \right)}\, dx$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=23$$$ y $$$f{\left(x \right)} = \cos^{3}{\left(35 x \right)}$$$:

$${\color{red}{\int{23 \cos^{3}{\left(35 x \right)} d x}}} = {\color{red}{\left(23 \int{\cos^{3}{\left(35 x \right)} d x}\right)}}$$

Sea $$$u=35 x$$$.

Entonces $$$du=\left(35 x\right)^{\prime }dx = 35 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{35}$$$.

La integral se convierte en

$$23 {\color{red}{\int{\cos^{3}{\left(35 x \right)} d x}}} = 23 {\color{red}{\int{\frac{\cos^{3}{\left(u \right)}}{35} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{35}$$$ y $$$f{\left(u \right)} = \cos^{3}{\left(u \right)}$$$:

$$23 {\color{red}{\int{\frac{\cos^{3}{\left(u \right)}}{35} d u}}} = 23 {\color{red}{\left(\frac{\int{\cos^{3}{\left(u \right)} d u}}{35}\right)}}$$

Extrae un coseno y expresa todo lo demás en términos del seno, utilizando la fórmula $$$\cos^2\left(\alpha \right)=-\sin^2\left(\alpha \right)+1$$$ con $$$\alpha= u $$$:

$$\frac{23 {\color{red}{\int{\cos^{3}{\left(u \right)} d u}}}}{35} = \frac{23 {\color{red}{\int{\left(1 - \sin^{2}{\left(u \right)}\right) \cos{\left(u \right)} d u}}}}{35}$$

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

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

Por lo tanto,

$$\frac{23 {\color{red}{\int{\left(1 - \sin^{2}{\left(u \right)}\right) \cos{\left(u \right)} d u}}}}{35} = \frac{23 {\color{red}{\int{\left(1 - v^{2}\right)d v}}}}{35}$$

Integra término a término:

$$\frac{23 {\color{red}{\int{\left(1 - v^{2}\right)d v}}}}{35} = \frac{23 {\color{red}{\left(\int{1 d v} - \int{v^{2} d v}\right)}}}{35}$$

Aplica la regla de la constante $$$\int c\, dv = c v$$$ con $$$c=1$$$:

$$- \frac{23 \int{v^{2} d v}}{35} + \frac{23 {\color{red}{\int{1 d v}}}}{35} = - \frac{23 \int{v^{2} d v}}{35} + \frac{23 {\color{red}{v}}}{35}$$

Aplica la regla de la potencia $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=2$$$:

$$\frac{23 v}{35} - \frac{23 {\color{red}{\int{v^{2} d v}}}}{35}=\frac{23 v}{35} - \frac{23 {\color{red}{\frac{v^{1 + 2}}{1 + 2}}}}{35}=\frac{23 v}{35} - \frac{23 {\color{red}{\left(\frac{v^{3}}{3}\right)}}}{35}$$

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

$$\frac{23 {\color{red}{v}}}{35} - \frac{23 {\color{red}{v}}^{3}}{105} = \frac{23 {\color{red}{\sin{\left(u \right)}}}}{35} - \frac{23 {\color{red}{\sin{\left(u \right)}}}^{3}}{105}$$

Recordemos que $$$u=35 x$$$:

$$\frac{23 \sin{\left({\color{red}{u}} \right)}}{35} - \frac{23 \sin^{3}{\left({\color{red}{u}} \right)}}{105} = \frac{23 \sin{\left({\color{red}{\left(35 x\right)}} \right)}}{35} - \frac{23 \sin^{3}{\left({\color{red}{\left(35 x\right)}} \right)}}{105}$$

Por lo tanto,

$$\int{23 \cos^{3}{\left(35 x \right)} d x} = - \frac{23 \sin^{3}{\left(35 x \right)}}{105} + \frac{23 \sin{\left(35 x \right)}}{35}$$

Simplificar:

$$\int{23 \cos^{3}{\left(35 x \right)} d x} = \frac{23 \left(3 - \sin^{2}{\left(35 x \right)}\right) \sin{\left(35 x \right)}}{105}$$

Añade la constante de integración:

$$\int{23 \cos^{3}{\left(35 x \right)} d x} = \frac{23 \left(3 - \sin^{2}{\left(35 x \right)}\right) \sin{\left(35 x \right)}}{105}+C$$

$$$\int 23 \cos^{3}{\left(35 x \right)}\, dx = \frac{23 \left(3 - \sin^{2}{\left(35 x \right)}\right) \sin{\left(35 x \right)}}{105} + C$$$A