Primitive of x squared by Cosine of a x

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Theorem

$\displaystyle \int x^2 \cos a x \rd x = \frac {2 x \cos a x} {a^2} + \left({\frac {x^2} a - \frac 2 {a^3} }\right) \sin a x + C$

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:

\(\displaystyle u\) \(=\) \(\displaystyle x^2\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\d u} {\d x}\) \(=\) \(\displaystyle 2 x\) Derivative of Power


and let:

\(\displaystyle \frac {\d v} {\d x}\) \(=\) \(\displaystyle \cos a x\)
\(\displaystyle \implies \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle \frac {\sin a x} a\) Primitive of $\cos a x$


Then:

\(\displaystyle \int x^2 \cos \left({a x}\right) \rd x\) \(=\) \(\displaystyle x^2 \left({\frac {\sin a x} a}\right) - \int \left({2 x \frac {\sin a x} a}\right) \rd x + C\) 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
\(\displaystyle \) \(=\) \(\displaystyle \frac {x^2 \sin a x} a - \frac 2 a \left({\frac {\sin a x} {a^2} - \frac {x \cos a x} a}\right) + C\) Primitive of $x \sin a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 x \cos a x} {a^2} + \left({\frac {x^2} a - \frac 2 {a^3} }\right) \sin a x + C\) simplification

$\blacksquare$


Also see


Sources