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
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