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Definite Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDefinite Integrals3 Marks
Question
Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\cos^2\text{x}\text{ dx}$
✓

Answer

Let $\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\cos^2\text{x}\text{ dx}$ Then,
$\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\cos^2\text{x}\text{ dx}$
$\Rightarrow\text{I}=\frac{1}{2}\int_{0}^\limits{\frac{\pi}{2}}(1+2\cos2\text{x})\text{dx}$ $[\because\cos2\text{x}=2\cos^2\text{x}-1\big]$
$\Rightarrow\text{I}=\Big[\frac{\pi}{2}+\frac{\sin2\text{x}}{4}\Big]^{\frac{\pi}{2}}_0$
$\Rightarrow\text{I}=\frac{\pi}{4}+0-0$
$\Rightarrow\text{I}=\frac{\pi}{4}$

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