How do i solve this question?

$\int \left({\cos}^{2} \frac{x}{\sin} x + \sin x + 1\right) \mathrm{dx}$

Mar 10, 2018

$I = - \ln \left(\csc \left(x\right) + \cot \left(x\right)\right) + x + C$

Explanation:

We want to solve

$I = \int {\cos}^{2} \frac{x}{\sin} \left(x\right) + \sin \left(x\right) + 1 \mathrm{dx}$

Rewrite the integrand

$I = \int \frac{1 - {\sin}^{2} \left(x\right)}{\sin} \left(x\right) + \sin \left(x\right) + 1 \mathrm{dx}$

$\textcolor{w h i t e}{I} = \int \csc \left(x\right) - \sin \left(x\right) + \sin \left(x\right) + 1 \mathrm{dx}$

$\textcolor{w h i t e}{I} = \int \csc \left(x\right) + 1 \mathrm{dx}$

$\textcolor{w h i t e}{I} = \int \csc \left(x\right) + \int 1 \mathrm{dx}$

We know the integral of cosecant as (otherwise look here )

${I}_{1} = \int \csc \left(x\right) \mathrm{dx} = - \ln \left(\left\mid \csc \left(x\right) + \cot \left(x\right) \right\mid\right) + C$

Thus

$I = - \ln \left(\left\mid \csc \left(x\right) + \cot \left(x\right) \right\mid\right) + x + C$