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Please solve RD Sharma class 12 chapter Indefinite Integrals exercise 18.9 question 37 maths textbook solution

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

Hint: Use substitution method to solve this integral.

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


        \begin{aligned} &\text { Let } I=\int \frac{(x+1) e^{x}}{\cos ^{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(x+1) e^{x} d x=d t \quad \text { then } \end{aligned}

        \Rightarrow I=\int \frac{1}{\cos ^{2} t} d t

        =\int \sec ^{2} t d t \quad\left[\because \frac{1}{\cos x}=\sec x\right]

        \begin{aligned} &=\tan t+c \\ &=\tan \left(x e^{x}\right)+c \quad\left[\because t=x e^{x}\right] \end{aligned}

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