Primitive of Fourth Power of Cosine Function

Theorem

$\ds \int \cos^4 x \rd x = \frac {3 x} 8 + \frac {\sin 2 x} 4 + \frac {\sin 4 x} {32} + C$

Proof

From Primitive of $\cos^4 a x$:

$\ds \int \cos^4 a x \rd x = \frac {3 x} 8 + \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$

The result follows by setting $a = 1$.

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $10$.