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need solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise 18.9 question 59

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Answer: \frac{1}{2} e^{x^{2}}+c

Hint: Use substitution method to solve this integral

Given: \int x\: e^{x^{2}} d x


        \text { Let } I=\int x\: e^{x^{2}} d x

        \begin{aligned} &\text { Put } x^{2}=t \Rightarrow 2 x d x=d t \Rightarrow d x=\frac{1}{2 x} d t \\ &\text { Then, } I=\int x \cdot e^{t} \cdot \frac{d t}{2 x}=\frac{1}{2} \int e^{t} d t=\frac{1}{2} e^{t}+c \end{aligned}                    \left[\because \int e^{x} d x=e^{x}+c\right]

                        =\frac{1}{2} e^{x^{2}}+c                                    \left[\because t=x^{2}\right]

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