first-make-a-substitution-and-then-use-integration-by-parts-to-evaluate-the-integral-displaystyle--4

Question

First make a substitution and then use integration by parts to evaluate the integral.

$ \displaystyle \int_{\sqrt{\frac{\pi}{2}}}^{\sqrt{\pi}} 	heta^3 \cos (	heta^2) d 	heta $

Wen Zheng · Accepted Answer

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Problem

First make a substitution and then use integratio…

Problem 39 Hard Difficulty

First make a substitution and then use integration by parts to evaluate the integral.

$ \displaystyle \int_{\sqrt{\frac{\pi}{2}}}^{\sqrt{\pi}} \theta^3 \cos (\theta^2) d \theta $

Answer

$\frac{1}{2}(\pi \sin \pi+\cos \pi)-\frac{1}{2}\left(\frac{\pi}{2} \sin \frac{\pi}{2}+\cos \frac{\pi}{2}\right)=\frac{1}{2}(\pi \cdot 0-1)-\frac{1}{2}\left(\frac{\pi}{2} \cdot 1+0\right)=-\frac{1}{2}-\frac{\pi}{4}$

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