Evaluate the following integrals: _ 0 ^ 4 (a^2 ^2x+b^2 ^2x )dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{4}}\big(\text{a}^2\cos^2\text{x}+\text{b}^2\sin^2\text{x}\big)\text{dx}$

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Answer

$\int_{0}^\limits{\frac{\pi}{4}}\big(\text{a}^2\cos^2\text{x}+\text{b}^2\sin^2\text{x}\big)\text{dx}$
$=\int_{0}^\limits{\frac{\pi}{4}}\Big[\text{a}^2\Big(\frac{1+\cos2\text{x}}{2}\Big)+\text{b}^2\Big(\frac{1-\cos2\text{x}}{2}\Big)\Big]\text{dx}$
$=\int_{0}^\limits{\frac{\pi}{4}}\Big[\Big(\frac{\text{a}^2+\text{b}^2}{2}\Big)+\Big(\frac{\text{a}^2-\text{b}^2}{2}\Big)\cos2\text{x}\Big]\text{dx}$
$=\Big(\frac{\text{a}^2+\text{b}^2}{2}\Big)\int_{0}^\limits{\frac{\pi}{4}}\text{dx}+\Big(\frac{\text{a}^2-\text{b}^2}{2}\Big)\int_{0}^\limits{\frac{\pi}{4}}\cos2\text{x }\text{dx}$
$=\Big(\frac{\text{a}^2+\text{b}^2}{2}\Big)\Big(\frac{\pi}{4}-0\Big)+\Big(\frac{\text{a}^2-\text{b}^2}{2}\Big)\Big(\sin\frac{\pi}{2}-\sin0\Big)$
$=\Big(\frac{\text{a}^2+\text{b}^2}{2}\Big)\frac{\pi}{4}+\Big(\frac{\text{a}^2-\text{b}^2}{2}\Big)(1-0)$
$=\big(\text{a}^2+\text{b}^2\big)\frac{\pi}{8}+\frac{1}{4}\big(\text{a}^2-\text{b}^2\big)$

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