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  • Please solve RD Sharma class 12 chapter 19 Definite Integrals exercise Fill in the blanks question 17 maths textbook solution

Please solve RD Sharma class 12 chapter 19 Definite Integrals exercise Fill in the blanks question 17 maths textbook solution

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Answer: 0

Hint: Use \frac{d}{d x} \sin x \text { and } \int x^{n} d x

Given: \int_{0}^{2 \pi} \cos ^{7} x \sin ^{4} x\; dx

Solution:  

\begin{aligned} &I=\int_{0}^{2 \pi} \cos ^{7} x \sin ^{4} x\; dx\\\\ &=\int_{0}^{2 \pi} \cos ^{6} x \sin ^{4} x \cdot \cos x\; dx \end{aligned}

\begin{aligned} &=\int_{0}^{2 \pi}\left(\cos ^{2} x\right)^{3} \sin ^{4} x \cdot \cos x\; dx \\\\ &=\int_{0}^{2 \pi}\left(1-\sin ^{2} x\right)^{3} \sin ^{4} x \cdot \cos x\; dx \end{aligned}

Put

        \begin{aligned} &\sin x=t \\\\ &\cos x\; d x=d t \end{aligned}

\begin{aligned} &I=\int_{0}^{2 \pi}\left(1-t^{2}\right)^{3} t^{4} d t \\\\ &=\int_{0}^{2 \pi}\left(1-t^{6}-3 t^{2}+3 t^{4}\right) t^{4} d t \end{aligned}

\begin{aligned} &=\int_{0}^{2 \pi}\left(t^{4}-t^{10}-3 t^{6}+3 t^{8}\right) d t \\\\ &=\left[\frac{t^{5}}{5}-\frac{t^{11}}{11}-\frac{3 t^{7}}{7}+\frac{3 t^{9}}{9}\right]_{0}^{2 \pi} \end{aligned}

Where t= sinx

=\left[\frac{\sin ^{5} x}{5}-\frac{\sin ^{11} x}{11}-\frac{3 \sin ^{7} x}{7}+\frac{3 \sin ^{9} x}{9}\right]_{0}^{2 \pi}

We know  \sin (2 \pi)=0

\begin{aligned} =& \frac{0}{5}-\frac{0}{11}-\frac{0}{7}-\frac{0}{9} \\\\ =& 0 \end{aligned}

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