Primitive of Square of Cosine Function/Proof 1

Theorem

$\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$

Proof

\(\ds \int \cos^2 x \rd x\)

\(=\)

\(\ds \int \paren {\frac {1 + \cos 2 x} 2} \rd x\)

Square of Cosine

\(\ds \)

\(=\)

\(\ds \int \frac 1 2 \rd x + \int \paren {\frac {\cos 2 x} 2} \rd x\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds \frac x 2 + C + \int \paren {\frac {\cos 2 x} 2} \rd x\)

Primitive of Constant

\(\ds \)

\(=\)

\(\ds \frac x 2 + C + \frac 1 2 \int \cos 2 x \rd x\)

Primitive of Constant Multiple of Function

\(\ds \)

\(=\)

\(\ds \frac x 2 + \frac 1 2 \paren {\frac {\sin 2 x} 2} + C\)

Primitive of Function of Constant Multiple and Primitive of Cosine Function

\(\ds \)

\(=\)

\(\ds \frac x 2 + \frac {\sin 2 x} 4 + C\)

Primitive of Function of Constant Multiple and Primitive of Cosine Function

$\blacksquare$