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Integration - Eg2c.tex
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\documentclass[10pt]{article} \usepackage{amssymb,amsmath} \begin{document} Eg2 (c) Show that $\int^{\pi}_{0}x \cos^2 x \: dx=\frac{\pi ^2}{4}$\\ \begin{align*} \int^{\pi}_{0}x \: dx &= \left[\frac{x^2}{2}\right]^{\pi}_0\\ &=\frac{{\pi}^2}{2}\\ \text{but} \quad \int^{\pi}_{0}x \: dx &= \int^{\pi}_{0}x \left(\sin^2 x + \cos^2 x\right) \: dx\\ =&\int^{\pi}_{0}x \left[\cos^2 \left(\frac{\pi}{2}-x\right) + \cos^2 x\right] \: dx\\ =& \int^{\pi}_{0}x \left[\cos^2 x + \cos^2 x\right] \: dx\\ =& 2\int^{\pi}_{0}x \cos^2 x \: dx\\ \\ \therefore \int^{\pi}_{0}x \cos^2 x \: dx =& \frac{1}{2} \int^{\pi}_{0}x \: dx\\ =&\frac{{\pi}^2}{4}\\ %&= \int^{\pi}_{0}x \left[\cos^2 \left(\frac{\pi}{2}-x\right) + \cos^2 x\right\right] \: dx\\ %&= \int^{\frac{\pi}{2}}_{0}x \left(\sin^2 x + \cos^2 x\right) \: dx %+ \int^{\pi}_{\frac{\pi}{2}}x \left(\sin^2 x + \cos^2 x\right) \: dx\\ %&= \int^{\frac{\pi}{2}}_{0}x \left[\cos^2 \left(\frac{\pi}{2}-x\right) + \cos^2 %x\right] \: dx %+ \int^{\pi}_{\frac{\pi}{2}}x \left(\sin^2 x + \cos^2 x\right) \: dx\\ \end{align*} \end{document}