$\int$ e^x(1 - cot x + cosec_² x) dx = ______________.

$I=\int e^x\left(1-\cot x+{\mathrm{cosec}}^{2}x\right)dx$

$\Rightarrow I=\int e^{x}dx+\int e^x\left(-\cot x+{\mathrm{cosec}}^{2}x\right)dx$

$\Rightarrow I=e^x+\int e^x\left(-\cot x+{\mathrm{cosec}}^{2}x\right)dx$

Let $f\left(x\right)=-\cot x$

$\Rightarrow f\text{'}\left(x\right)=-\left(-{\mathrm{cosec}}^{2}x\right)={\mathrm{cosec}}^{2}x$

$\therefore I=e^x+e^x\left(-\cot x\right)+C$ $\left[\int e^x\left[f\left(x\right)+f\text{'}\left(x\right)\right]dx=e^{x}f\left(x\right)+C\right]$

$\Rightarrow I=e^x\left(1-\cot x\right)+C$


$\int$ e^x(1 − cot x + cosec_² x) dx = $\underline{e^x\left(1-\cot x\right)+C}$.