# Primitive of x by Sine of a x

## Theorem

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

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:

 $\displaystyle u$ $=$ $\displaystyle x$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d u} {\d x}$ $=$ $\displaystyle 1$ Derivative of Identity Function

and let:

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

Then:

 $\displaystyle \int x \map \sin {a x} \rd x$ $=$ $\displaystyle x \paren {-\frac {\cos a x} a} - \int \paren {-\frac {\cos a x} a} \times 1 \rd x + C$ Integration by Parts $\displaystyle$ $=$ $\displaystyle -\frac {x \cos a x} a + \frac 1 a \int \cos a x \rd x + C$ Linear Combination of Primitives $\displaystyle$ $=$ $\displaystyle -\frac {x \cos a x} a + \frac 1 a \paren {\frac {\sin a x} a} + C$ Primitive of $\cos a x$ $\displaystyle$ $=$ $\displaystyle \frac {\sin a x} {a^2} - \frac {x \cos a x} a + C$ simplification

$\blacksquare$