Integral of $$$\sin{\left(x \right)} \sin{\left(3 x \right)}$$$

Find $$$\int \sin{\left(x \right)} \sin{\left(3 x \right)}\, dx$$$.

Rewrite integrand using the formula $$$\sin\left(\alpha \right)\sin\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)-\frac{1}{2} \cos\left(\alpha+\beta \right)$$$ with $$$\alpha=x$$$ and $$$\beta=3 x$$$:

$${\color{red}{\int{\sin{\left(x \right)} \sin{\left(3 x \right)} d x}}} = {\color{red}{\int{\left(\frac{\cos{\left(2 x \right)}}{2} - \frac{\cos{\left(4 x \right)}}{2}\right)d x}}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(x \right)} = \cos{\left(2 x \right)} - \cos{\left(4 x \right)}$$$:

$${\color{red}{\int{\left(\frac{\cos{\left(2 x \right)}}{2} - \frac{\cos{\left(4 x \right)}}{2}\right)d x}}} = {\color{red}{\left(\frac{\int{\left(\cos{\left(2 x \right)} - \cos{\left(4 x \right)}\right)d x}}{2}\right)}}$$

Integrate term by term:

$$\frac{{\color{red}{\int{\left(\cos{\left(2 x \right)} - \cos{\left(4 x \right)}\right)d x}}}}{2} = \frac{{\color{red}{\left(\int{\cos{\left(2 x \right)} d x} - \int{\cos{\left(4 x \right)} d x}\right)}}}{2}$$

Let $$$u=4 x$$$.

Then $$$du=\left(4 x\right)^{\prime }dx = 4 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{4}$$$.

Therefore,

$$\frac{\int{\cos{\left(2 x \right)} d x}}{2} - \frac{{\color{red}{\int{\cos{\left(4 x \right)} d x}}}}{2} = \frac{\int{\cos{\left(2 x \right)} d x}}{2} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{2}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{4}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{\int{\cos{\left(2 x \right)} d x}}{2} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{2} = \frac{\int{\cos{\left(2 x \right)} d x}}{2} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}}{2}$$

The integral of the cosine is $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$\frac{\int{\cos{\left(2 x \right)} d x}}{2} - \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{8} = \frac{\int{\cos{\left(2 x \right)} d x}}{2} - \frac{{\color{red}{\sin{\left(u \right)}}}}{8}$$

Recall that $$$u=4 x$$$:

$$\frac{\int{\cos{\left(2 x \right)} d x}}{2} - \frac{\sin{\left({\color{red}{u}} \right)}}{8} = \frac{\int{\cos{\left(2 x \right)} d x}}{2} - \frac{\sin{\left({\color{red}{\left(4 x\right)}} \right)}}{8}$$

Let $$$u=2 x$$$.

Then $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{2}$$$.

The integral can be rewritten as

$$- \frac{\sin{\left(4 x \right)}}{8} + \frac{{\color{red}{\int{\cos{\left(2 x \right)} d x}}}}{2} = - \frac{\sin{\left(4 x \right)}}{8} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{2}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$- \frac{\sin{\left(4 x \right)}}{8} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{2} = - \frac{\sin{\left(4 x \right)}}{8} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}}{2}$$

The integral of the cosine is $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$- \frac{\sin{\left(4 x \right)}}{8} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{4} = - \frac{\sin{\left(4 x \right)}}{8} + \frac{{\color{red}{\sin{\left(u \right)}}}}{4}$$

Recall that $$$u=2 x$$$:

$$- \frac{\sin{\left(4 x \right)}}{8} + \frac{\sin{\left({\color{red}{u}} \right)}}{4} = - \frac{\sin{\left(4 x \right)}}{8} + \frac{\sin{\left({\color{red}{\left(2 x\right)}} \right)}}{4}$$

Therefore,

$$\int{\sin{\left(x \right)} \sin{\left(3 x \right)} d x} = \frac{\sin{\left(2 x \right)}}{4} - \frac{\sin{\left(4 x \right)}}{8}$$

Simplify:

$$\int{\sin{\left(x \right)} \sin{\left(3 x \right)} d x} = \sin^{3}{\left(x \right)} \cos{\left(x \right)}$$

Add the constant of integration:

$$\int{\sin{\left(x \right)} \sin{\left(3 x \right)} d x} = \sin^{3}{\left(x \right)} \cos{\left(x \right)}+C$$

$$$\int \sin{\left(x \right)} \sin{\left(3 x \right)}\, dx = \sin^{3}{\left(x \right)} \cos{\left(x \right)} + C$$$A