MHT CET · Maths · Indefinite Integration
\(\int \mathrm{e}^x\left(1-\cot x+\cot ^2 x\right) \mathrm{d} x=\)
- A \(\mathrm{e}^x \cdot \cot x+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
- B \(\mathrm{e}^x \cdot \operatorname{cosec} x+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
- C \(-\mathrm{e}^x \cdot \cot x+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
- D \(-\mathrm{e}^x \cdot \operatorname{cosec} x+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
Answer & Solution
Correct Answer
(C) \(-\mathrm{e}^x \cdot \cot x+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \int \mathrm{e}^x\left(1-\cot x+\cot ^2 x\right) \mathrm{d} x \\ & =\int \mathrm{e}^x\left(1+\cot ^2 x-\cot x\right) \mathrm{d} x \\ & =\int \mathrm{e}^x\left(-\cot x+\operatorname{cosec}^2 x\right) \mathrm{d} x \\ & =\mathrm{e}^x(-\cot x)+\mathrm{c} \\ & \quad \cdots\left[\because \int \mathrm{e}^x\left[\mathrm{f}(x)+\mathrm{f}^{\prime}(x)\right] \mathrm{d} x=\mathrm{e}^x \mathrm{f}(x)+\mathrm{c}\right] \\ & =-\mathrm{e}^x \cdot \cot x+\mathrm{c}\end{aligned}\)
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