Primitive of Square of Cosine Function

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Theorem

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

where $C$ is an arbitrary constant.


Corollary

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

where $C$ is an arbitrary constant.


Proof 1

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

$\blacksquare$


Proof 2

\(\ds I_n\) \(=\) \(\ds \int \cos^n x \rd x\)
\(\ds \) \(=\) \(\ds \dfrac {\cos^{n - 1} x \sin x} n + \dfrac {n - 1} n I_{n-2}\) 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

$\blacksquare$


Proof 3

\(\ds \int \cos^2 x \rd x\) \(=\) \(\ds \frac 1 4 \int \paren {e^{i x} + e^{-i x} }^2 \rd x\) 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

$\blacksquare$


Also presented as

Some sources present this as:

$\ds \int \cos^2 x \rd x = \frac 1 2 \paren {x + \frac {\sin 2 x} 2} + C$


Sources