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Provide solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Very short answer type question 6

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Hint: You must know the integration rules of trigonometric function with its limits

Given: \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} x \cos ^{2} x \; d x

Solution:  I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} x \cos ^{2} x \; d x

\begin{aligned} &f(x)=x \cos ^{2} x \\\\ &f(-x)=(-x) \cos ^{2}(-x) \\\\ &=-x \cos ^{2} x \\\\ &=-f(x) \end{aligned}

Hence, f(x) is an odd function.

Since,\int_{-a}^{a} f(x) d x=0    if f(x) is an odd

\therefore \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} x \cos ^{2} x\; d x=0

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