Primitive of Square of Cosine Function/Proof 3

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

$\blacksquare$