Mathematics

# Evaluate the following integrals :$\displaystyle\int_{0}^{\pi}x\sin x\cos^{4}x\ dx$

##### SOLUTION

Let $I=\displaystyle\int_{0}^{\pi}x\sin x\cos^{4}x\ dx$.

Then, $I=\displaystyle\int_{0}^{\pi}(\pi-x)\sin (\pi-x)\cos^{4}(\pi-x)dx$

$\displaystyle\int_{0}^{\pi}(\pi-x)\sin x\cos^{4}x\ dx$

Adding $(1)$ and $(2)$, we get

$\therefore 2I=-\pi\left[\dfrac{t^{5}}{5}\right]_{1}^{-1}$

$=-\pi\left(-\dfrac{1}{5}-\dfrac{1}{5}\right)$

$=\dfrac{2\pi}{5}$

$\Rightarrow I=\dfrac{\pi}{5}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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