Question

\[\int e^x \left( 1 - \cot x + \cot^2 x \right) dx =\]

Options

•  e^x cot x + C

• −e^x cot x + C

•  e^x cosec x + C

• −e^x cosec x + C

Answer 1

−e^x cot x + C 

\[\text{Let }I = \int e^x \left( 1 - \cot x + \cot^2 x \right)dx\]

\[ = \int e^x \left( {cosec}^2 x - \cot x \right)dx\]

\[\text{As we know that }\int e\left\{ f\left( x \right) + f' {}^x \left( x \right) \right\} = e^x f\left( x \right) + C\]

\[ \therefore I = - e^x \cot x + C\]