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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS2 Marks
Question
By using the properties of definite integral, evaluate the integral in Exercise:
$\int\limits_{0}^{\frac{\pi}{2}}\cos^{2}\text{x}\ \text{dx}$
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Answer

$\text{Let}\ \text{I}=\int\limits_{0}^{\frac{\pi}{2}}\cos^{2}\text{x}\ \text{dx}$
$=\int\limits_{0}^{\frac{\pi}{2}}\cos^{2}\bigg(\frac{\pi}{2}-\text{x}\bigg)\text{dx}\ \ \ \bigg[\because\int\limits_{0}^{\text{a}}\text{(x)}\text{dx}=\int\limits_{0}^{\text{a}}\text{f}(\text{a}-\text{x})\text{dx}=\bigg]$
$\Rightarrow\ \int\limits_{0}^{\frac{\pi}{2}}\sin^{2}\text{x}\ \text{dx}$
Adding eq.(i) and (ii),
$21=\int\limits_{0}^{\frac{\pi}{2}}(\cos^{2}\text{x}+\sin^{2}\text{x})\text{dx}=\int\limits_{0}^{\frac{\pi}{2}}1\ \text{dx}=\bigg(\text{x}^{\frac{\pi}{2}}_{0}\bigg)$
$\Rightarrow\ 21=\frac{\pi}{2}\ \ \ \Rightarrow\ \ \text{I}=\frac{\pi}{4}$

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