Question

Evaluate : $\int e^x\left[\frac{\cos x-\sin x}{{\sin }^{2}x}\right]dx$.

Answer 1

We need to evaluate $\int e^x\left[\frac{cosx-sinx}{si{n}^{2}x}\right]dx$

$\int e^x\left[\frac{cosx-sinx}{si{n}^{2}x}\right]dx=\int e^x[\frac{cosx}{si{n}^{2}x}-\frac{sinx}{si{n}^{2}x}]dx$

$=\int e^x[\frac{1}{sinx}\cdot \frac{cosx}{sinx}-\frac{1}{sinx}]dx$

$=\int e^x[cosecx\cdot cotx-cosecx]dx$

Let $f\left(x\right)=-cosecx\Rightarrow f^{\prime }\left(x\right)=cosecxcotx$

$=\int e^x\left[f\right(x)+f^{\prime }(x\left)\right]dx$

$=e^{x}f\left(x\right)+c$

$=e^x(-cosecx)+c$

$\therefore \int e^x\left[\frac{cosx-sinx}{si{n}^{2}x}\right]dx=-e^{x}cosecx+c$