# Primitive of x squared by Cosine of a x

## Theorem

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

where $C$ is an arbitrary constant.

## Proof

With a view to expressing the primitive in the form:

$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

 $\displaystyle u$ $=$ $\displaystyle x^2$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d u} {\d x}$ $=$ $\displaystyle 2 x$ Derivative of Power

and let:

 $\displaystyle \frac {\d v} {\d x}$ $=$ $\displaystyle \cos a x$ $\displaystyle \leadsto \ \$ $\displaystyle v$ $=$ $\displaystyle \frac {\sin a x} a$ Primitive of $\cos a x$

Then:

 $\displaystyle \int x^2 \map \cos {a x} \rd x$ $=$ $\displaystyle x^2 \paren {\frac {\sin a x} a} - \int \paren {2 x \frac {\sin a x} a} \rd x + C$ Integration by Parts $\displaystyle$ $=$ $\displaystyle \frac {x^2 \sin a x} a - \frac 2 a \int x \sin a x \rd x + C$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \frac {x^2 \sin a x} a - \frac 2 a \paren {\frac {\sin a x} {a^2} - \frac {x \cos a x} a} + C$ Primitive of $x \sin a x$ $\displaystyle$ $=$ $\displaystyle \frac {2 x \cos a x} {a^2} + \paren {\frac {x^2} a - \frac 2 {a^3} } \sin a x + C$ simplification

$\blacksquare$