Definite Integral from 0 to Half Pi of Square of Cosine x/Proof 1
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Theorem
- $\ds \int_0^{\frac \pi 2} \cos^2 x \rd x = \frac \pi 4$
Proof
\(\ds \int \cos^2 x \rd x\) | \(=\) | \(\ds \frac x 2 + \frac {\sin 2 x} 4 + C\) | Primitive of $\cos^2 x$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int_0^{\frac \pi 2} \cos^2 x \rd x\) | \(=\) | \(\ds \intlimits {\frac x 2 + \frac {\sin 2 x} 4} 0 {\frac \pi 2}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac \pi 4 + \frac {\sin \pi} 4} - \paren {\frac 0 2 + \frac {\sin 0} 4}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 4\) | Sine of Multiple of Pi and simplifying |
$\blacksquare$