Integral of $$$\sin{\left(44 x \right)} \sin{\left(\cos{\left(44 x \right)} \right)}$$$

Find $$$\int \sin{\left(44 x \right)} \sin{\left(\cos{\left(44 x \right)} \right)}\, dx$$$.

Let $$$u=44 x$$$.

Then $$$du=\left(44 x\right)^{\prime }dx = 44 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{44}$$$.

The integral can be rewritten as

$${\color{red}{\int{\sin{\left(44 x \right)} \sin{\left(\cos{\left(44 x \right)} \right)} d x}}} = {\color{red}{\int{\frac{\sin{\left(u \right)} \sin{\left(\cos{\left(u \right)} \right)}}{44} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{44}$$$ and $$$f{\left(u \right)} = \sin{\left(u \right)} \sin{\left(\cos{\left(u \right)} \right)}$$$:

$${\color{red}{\int{\frac{\sin{\left(u \right)} \sin{\left(\cos{\left(u \right)} \right)}}{44} d u}}} = {\color{red}{\left(\frac{\int{\sin{\left(u \right)} \sin{\left(\cos{\left(u \right)} \right)} d u}}{44}\right)}}$$

Let $$$v=\cos{\left(u \right)}$$$.

Then $$$dv=\left(\cos{\left(u \right)}\right)^{\prime }du = - \sin{\left(u \right)} du$$$ (steps can be seen »), and we have that $$$\sin{\left(u \right)} du = - dv$$$.

So,

$$\frac{{\color{red}{\int{\sin{\left(u \right)} \sin{\left(\cos{\left(u \right)} \right)} d u}}}}{44} = \frac{{\color{red}{\int{\left(- \sin{\left(v \right)}\right)d v}}}}{44}$$

Apply the constant multiple rule $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ with $$$c=-1$$$ and $$$f{\left(v \right)} = \sin{\left(v \right)}$$$:

$$\frac{{\color{red}{\int{\left(- \sin{\left(v \right)}\right)d v}}}}{44} = \frac{{\color{red}{\left(- \int{\sin{\left(v \right)} d v}\right)}}}{44}$$

The integral of the sine is $$$\int{\sin{\left(v \right)} d v} = - \cos{\left(v \right)}$$$:

$$- \frac{{\color{red}{\int{\sin{\left(v \right)} d v}}}}{44} = - \frac{{\color{red}{\left(- \cos{\left(v \right)}\right)}}}{44}$$

Recall that $$$v=\cos{\left(u \right)}$$$:

$$\frac{\cos{\left({\color{red}{v}} \right)}}{44} = \frac{\cos{\left({\color{red}{\cos{\left(u \right)}}} \right)}}{44}$$

Recall that $$$u=44 x$$$:

$$\frac{\cos{\left(\cos{\left({\color{red}{u}} \right)} \right)}}{44} = \frac{\cos{\left(\cos{\left({\color{red}{\left(44 x\right)}} \right)} \right)}}{44}$$

Therefore,

$$\int{\sin{\left(44 x \right)} \sin{\left(\cos{\left(44 x \right)} \right)} d x} = \frac{\cos{\left(\cos{\left(44 x \right)} \right)}}{44}$$

Add the constant of integration:

$$\int{\sin{\left(44 x \right)} \sin{\left(\cos{\left(44 x \right)} \right)} d x} = \frac{\cos{\left(\cos{\left(44 x \right)} \right)}}{44}+C$$

$$$\int \sin{\left(44 x \right)} \sin{\left(\cos{\left(44 x \right)} \right)}\, dx = \frac{\cos{\left(\cos{\left(44 x \right)} \right)}}{44} + C$$$A