Primitive of Square of Cosine Function/Proof 3
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Theorem
- $\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$
Proof
\(\ds \int \cos^2 x \rd x\) | \(=\) | \(\ds \frac 1 4 \int \paren {e^{i x} + e^{-i x} }^2 \rd x\) | Euler's Cosine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 4 \int \paren {e^{2 i x} + 2 + e^{-2 i x} } \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 4 \paren {\frac{e^{2 i x} - e^{-2 i x} } {2 i} + 2 x} + C\) | Primitive of $e^{a x}$, Primitive of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin 2 x} 4 + \frac x 2 + C\) | Euler's Sine Identity |
$\blacksquare$