Primitive of x squared by Cosine of a x

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

where $C$ is an arbitrary constant.


With a view to expressing the primitive in the form:

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


\(\displaystyle u\) \(=\) \(\displaystyle x^2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d u} {\d x}\) \(=\) \(\displaystyle 2 x\) 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 $\cos a x$


\(\displaystyle \int x^2 \map \cos {a x} \rd x\) \(=\) \(\displaystyle x^2 \paren {\frac {\sin a x} a} - \int \paren {2 x \frac {\sin a x} a} \rd x + C\) 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


Also see