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Please solve RD Sharma class 12 chapter Indefinite Integrals exercise 18.9 question 33 maths textbook solution

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Answer: \frac{1}{4} \sin \left(x^{4}\right)+c

Hint:Use substitution method to solve this integral.

Given:   \int x^{3} \cos x^{4} d x


        \begin{aligned} &\text { Let } I=\int x^{3} \cos x^{4} d x \\ &\text { Put } x^{4}=t \Rightarrow 4 x^{3} d x=d t \Rightarrow d x=\frac{d t}{4 x^{3}} \text { then } \end{aligned}

        \Rightarrow I=\int x^{3} \cos t \frac{d t}{4 x^{3}}=\frac{1}{4} \int \cos t \; d t

        =\frac{1}{4} \int \sin t \; d t \quad\left[\because \int \cos x \; d x=\sin x+c\right]

        =\frac{1}{4} \sin \left(x^{4}\right)+c \quad\left[\because t=x^{4}\right]


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