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  • Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise Multiple Choice Questions Question 14 Maths Textbook Solution.

Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise Multiple Choice Questions Question 14 Maths Textbook Solution.

Answers (1)

Answer:

-e^{x} \cot x+C

Given:

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

Hint:

Using integration by parts

Explanation:

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

          \begin{aligned} &=\int e^{x}\left[\left(1+\cot ^{2} x\right)-\cot x\right] d x \\ &=\int e^{x}\left(\operatorname{cosec}^{2} x-\cot x\right) d x \end{aligned}                                                \left[\because 1+\cot ^{2} \theta=\cos e c^{2} \theta\right]

          \begin{aligned} &=-\int e^{x}\left(\cot x-\cos e c^{2} x\right) d x \\ =&-\int e^{x} \cot x d x+\int e^{x} \operatorname{cosec}^{2} x d x \\ =&-\left[\cot x \cdot e^{x}-\int\left(-\cos e c^{2} x\right) e^{x} d x\right]+\int e^{x} \cos e c^{2} x d x \ldots .\left\{\int u \cdot v d x=u \int v d x-\int \frac{d u}{d x} \int v d x d x\right\} \\ =&-\cot x \cdot e^{x}-\int e^{x} \cos e c^{2} x d x+\int e^{x} \cos e c^{2} x d x \\ =&-\cot x e^{x}+C \quad \text { [Using integration by parts ] } \end{aligned}

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