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provide solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise  18.9 question 38

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Answer: \frac{1}{3} \sin \left(e^{x^{3}}\right)+c

Hint: Use substitution method to solve this integral.

Given:   \int x^{2} e^{x^{3}} \cos \left(e^{x^{3}}\right) d x


        \begin{aligned} &\text { Let } I=\int x^{2} e^{x^{3}} \cos \left(e^{x^{3}}\right) d x \\ &\text { Put } e^{x^{3}}=t \Rightarrow e^{x^{3}} \cdot 3 x^{2} d x=d t \\ &\Rightarrow d x=\frac{d t}{e^{x^{3}} \cdot 3 x^{2}} \text { then } \end{aligned}

        \Rightarrow I=\int e^{x^{3}} \cdot x^{2} \cdot \cos t \frac{d t}{e^{x^{3}} .3 x^{2}}=\frac{1}{3} \int \cos t\; d t

        =\frac{1}{3} \sin t \; d t \quad\left[\because \int \cos x \; d x=\sin x+c\right]

        =\frac{1}{3} \sin \left(e^{x^{3}}\right)+c \quad\left[\because t=e^{x^{3}}\right]

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