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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 $

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

Integration Techniques

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Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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