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Provide Solutio for RD Sharma Class 12 Chapter Indefinite Integrals Exercise 18.12 Question 4

Answers (1)

Answer:-   \frac{1}{6} \sin ^{6} x+C

Hint: - Use substitution method to solve this integral.

Given:-   \int \sin ^{5} x \cdot \cos x d x

Solution: - Let   \int \sin ^{5} x \cdot \cos x d x


\begin{aligned} \operatorname{Sin} x &=t \Rightarrow \cos x d x=d t, \text { then } \\ \quad I &=\int t^{5} \cdot d t \\ \quad &=\frac{t^{5}+1}{5+1}+C \quad\quad\quad\quad\quad\quad\quad\quad\left[\therefore \int x^{n} d x=\frac{x^{n+1}}{n+1}+C\right] \end{aligned}
           \begin{aligned} &=\frac{t^{6}}{6}+C \\ &=\frac{\sin ^{6} x}{6}+C \quad\quad\quad\quad\quad\quad\quad\quad[\because t=\sin x] \end{aligned}

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