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Explain solution RD Sharma class 12 Chapter 19 Definite Integrals Exercise Revision Exercise question 34

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Answer:  0

Hint: When the value of  f\left ( x \right )  is odd then answer is zero

Given:   \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{9} x d x

Solution: \begin{aligned} &f(x)=\sin ^{9} x \\ & \end{aligned}

               x \rightarrow-x                              \left[\int_{-a}^{a} f(x) d x=0, f(x)->o d d\right]

f(-x)=\sin ^{9}(-x)=>(-\sin x)^{9}

              =-\sin ^{9} x

f(-x)=-f(x) \Rightarrow \mathrm{f}(\mathrm{x}) \text { is odd }

\int_{-a}^{a} f(x) d x=0, \text { when } \mathrm{f}(\mathrm{x}) \text { is odd }

\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{9} x d x=0

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