Primitive of x cubed by Cosine of a x

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int x^3 \map \cos {a x} \rd x = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \cos a x + \paren {\frac {x^3} a - \frac {6 x} {a^3} } \sin a x + C$

where $C$ is an arbitrary constant.


Proof

With a view to expressing the primitive in the form:

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

let:

\(\ds u\) \(=\) \(\ds x^3\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds 3 x^2\) Derivative of Power


and let:

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


Then:

\(\ds \int x^3 \map \cos {a x} \rd x\) \(=\) \(\ds x^3 \paren {\frac {\sin a x} a} - \int \paren {3 x^2 \frac {\sin a x} a} \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac {x^3 \sin a x} a - \frac 3 a \int x^2 \sin a x \rd x + C\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac {x^3 \sin a x} a - \frac 3 a \paren {\frac {2 x \sin a x} {a^2} + \paren {\frac 2 {a^3} - \frac {x^2} a} \cos a x} + C\) Primitive of $x^2 \sin a x$
\(\ds \) \(=\) \(\ds \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \cos a x + \paren {\frac {x^3} a - \frac {6 x} {a^3} } \sin a x + C\) simplification

$\blacksquare$


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Primitive_of_x_cubed_by_Cosine_of_a_x&oldid=598271"