Primitive of x fourth by Cosine of a x
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Theorem
- $\ds \int x^4 \cos a x \rd x = \frac {\sin a x} a x^4 + \frac {4 \cos a x} {a^2} x^3 - \frac {12 \sin a x} {a^3} x^2 - \frac {24 \cos a x} {a^4} x + \frac {24 \sin a x} {a^5} + C$
where $C$ is an arbitrary constant.
Proof
| \(\ds \int x^4 \cos a x \rd x\) | \(=\) | \(\ds \frac {\sin a x} a x^4 + \frac {4 \cos a x} {a^2} x^{4 - 1} - \frac {4 \paren {4 - 1} } {a^2} \int x^{4 - 2} \cos a x \rd x + C\) | Primitive of $x^n \cos a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin a x} a x^4 + \frac {4 \cos a x} {a^2} x^3 - \frac {12} {a^2} \int x^2 \cos a x \rd x + C\) | Simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin a x} a x^4 + \frac {4 \cos a x} {a^2} x^3 - \frac {12} {a^2} \paren {\frac {2 x \cos a x} {a^2} + \paren {\frac {x^2} a - \frac 2 {a^3} } \sin a x} + C\) | Primitive of $x^2 \cos a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin a x} a x^4 + \frac {4 \cos a x} {a^2} x^3 - \frac {12 \sin a x} {a^3} x^2 - \frac {24 \cos a x} {a^4} x + \frac {24 \sin a x} {a^5} + C\) | simplifying |
$\blacksquare$