Integral of $$$\ln\left(x \sin{\left(c \right)}\right)$$$ with respect to $$$x$$$

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

Let $$$u=x \sin{\left(c \right)}$$$.

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

The integral can be rewritten as

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

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

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

For the integral $$$\int{\ln{\left(u \right)} d u}$$$, use integration by parts $$$\int \operatorname{m} \operatorname{dv} = \operatorname{m}\operatorname{v} - \int \operatorname{v} \operatorname{dm}$$$.

Let $$$\operatorname{m}=\ln{\left(u \right)}$$$ and $$$\operatorname{dv}=du$$$.

Then $$$\operatorname{dm}=\left(\ln{\left(u \right)}\right)^{\prime }du=\frac{du}{u}$$$ (steps can be seen ») and $$$\operatorname{v}=\int{1 d u}=u$$$ (steps can be seen »).

So,

$$\frac{{\color{red}{\int{\ln{\left(u \right)} d u}}}}{\sin{\left(c \right)}}=\frac{{\color{red}{\left(\ln{\left(u \right)} \cdot u-\int{u \cdot \frac{1}{u} d u}\right)}}}{\sin{\left(c \right)}}=\frac{{\color{red}{\left(u \ln{\left(u \right)} - \int{1 d u}\right)}}}{\sin{\left(c \right)}}$$

Apply the constant rule $$$\int c\, du = c u$$$ with $$$c=1$$$:

$$\frac{u \ln{\left(u \right)} - {\color{red}{\int{1 d u}}}}{\sin{\left(c \right)}} = \frac{u \ln{\left(u \right)} - {\color{red}{u}}}{\sin{\left(c \right)}}$$

Recall that $$$u=x \sin{\left(c \right)}$$$:

$$\frac{- {\color{red}{u}} + {\color{red}{u}} \ln{\left({\color{red}{u}} \right)}}{\sin{\left(c \right)}} = \frac{- {\color{red}{x \sin{\left(c \right)}}} + {\color{red}{x \sin{\left(c \right)}}} \ln{\left({\color{red}{x \sin{\left(c \right)}}} \right)}}{\sin{\left(c \right)}}$$

Therefore,

$$\int{\ln{\left(x \sin{\left(c \right)} \right)} d x} = \frac{x \ln{\left(x \sin{\left(c \right)} \right)} \sin{\left(c \right)} - x \sin{\left(c \right)}}{\sin{\left(c \right)}}$$

Simplify:

$$\int{\ln{\left(x \sin{\left(c \right)} \right)} d x} = x \left(\ln{\left(x \sin{\left(c \right)} \right)} - 1\right)$$

Add the constant of integration:

$$\int{\ln{\left(x \sin{\left(c \right)} \right)} d x} = x \left(\ln{\left(x \sin{\left(c \right)} \right)} - 1\right)+C$$

$$$\int \ln\left(x \sin{\left(c \right)}\right)\, dx = x \left(\ln\left(x \sin{\left(c \right)}\right) - 1\right) + C$$$A