Primitive of x squared by Cosine of x

Theorem

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

Proof

From Primitive of $x^2 \cos a x$:

$\ds \int x^2 \cos a x \rd x = \frac {2 x \cos a x} {a^2} + \paren {\frac {x^2} a - \frac 2 {a^3} } \sin a x + C$

The result follows on setting $a = 1$.

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $23$.