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explain solution RD Sharma class 12 chapter Indefinite Integrals exercise 18.9 question 36 maths

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

Hint: Use substitution method to solve this integral.

Given:   \int x^{3} \sin \left(x^{4}+1\right) d x


        \begin{aligned} &\text { Let } I=\int x^{3} \sin \left(x^{4}+1\right) d x \\ &\text { Put } x^{4}+1=t \Rightarrow 4 x^{3} d x=d t \\ &\Rightarrow d x=\frac{d t}{4 x^{3}} \text { then } \end{aligned}

        \begin{aligned} \Rightarrow I &=\int x^{3} \sin t \frac{d t}{4 x^{3}}=\frac{1}{4} \int \sin t\; d t \\ &=-\frac{\cos t}{4}+c \end{aligned}                                \left[\because \int \sin x\; d x=-\cos x+c\right]

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


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