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explain solution RD Sharma class 12 chapter Indefinite Integrals exercise 18.9 question 48 maths

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Answer: -\cot \left(x e^{x}\right)+c

Hint: Use substitution method to solve this integral.

Given:   \int \frac{(x+1) e^{x}}{\sin ^{2}\left(x e^{x}\right)} d x


        \begin{aligned} &\text { Let } I=\int \frac{(x+1) e^{x}}{\sin ^{2}\left(x e^{x}\right)} d x \\ &\text { Put } x e^{x}=t \Rightarrow\left(x e^{x}+1 . e^{x}\right) d x=d t \\ &\Rightarrow e^{x}(x+1) d x=d t \quad \text { then } \end{aligned}

        I=\int \frac{1}{\sin ^{2}(t)} d t=\int \operatorname{cosec}^{2} t \; d t \quad\left[\because \frac{1}{\sin x}=\operatorname{cosec\; x}\right]

            =[-\cot t]+\mathrm{c} \quad\left[\because \int \operatorname{cosec}^{2} x \; d x=-\cot x+c\right]

            =-\cot \left(x e^{x}\right)+c \quad\left[\because t=x e^{x}\right]

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