Primitive of Square of Cosine Function/Proof 2
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Theorem
- $\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$
Proof
\(\ds I_n\) | \(=\) | \(\ds \int \cos^n x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\cos^{n - 1} x \sin x} n + \dfrac {n - 1} n I_{n-2}\) | Reduction Formula for Integral of Power of Cosine | |||||||||||
\(\ds I_0\) | \(=\) | \(\ds \int \left({\cos x}\right)^0 \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x + C\) | Primitive of Constant | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds I_2\) | \(=\) | \(\ds \frac {\cos x \sin x} 2 + \frac x 2 + \frac C 2\) | setting $n = 2$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin 2 x} 4 + \frac x 2 + C'\) | Double Angle Formula for Sine |
$\blacksquare$