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

$\int \mathrm{e}^x\left(1-\cot x+\cot ^2 x\right) \mathrm{d} x=$
  1. $\mathrm{e}^x \cdot \cot x+\mathrm{c}$, where $\mathrm{c}$ is a constant of integration.
  2. $\mathrm{e}^x \cdot \operatorname{cosec} x+\mathrm{c}$, where $\mathrm{c}$ is a constant of integration.
  3. $-\mathrm{e}^x \cdot \cot x+\mathrm{c}$, where $\mathrm{c}$ is a constant of integration.
  4. $-\mathrm{e}^x \cdot \operatorname{cosec} x+\mathrm{c}$, where $\mathrm{c}$ is a constant of integration.

Solution

$\begin{aligned} & \int \mathrm{e}^x\left(1-\cot x+\cot ^2 x\right) \mathrm{d} x \\ & =\int \mathrm{e}^x\left(1+\cot ^2 x-\cot x\right) \mathrm{d} x \\ & =\int \mathrm{e}^x\left(-\cot x+\operatorname{cosec}^2 x\right) \mathrm{d} x \\ & =\mathrm{e}^x(-\cot x)+\mathrm{c} \\ & \quad \cdots\left[\because \int \mathrm{e}^x\left[\mathrm{f}(x)+\mathrm{f}^{\prime}(x)\right] \mathrm{d} x=\mathrm{e}^x \mathrm{f}(x)+\mathrm{c}\right] \\ & =-\mathrm{e}^x \cdot \cot x+\mathrm{c}\end{aligned}$

Asked in: MHT CET 2023 (10 May Shift 2)

Practice more Indefinite Integration questions on Aicharya