Primitive of x cubed by Sine of a x
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Theorem
- $\ds \int x^3 \sin a x \rd x = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \sin a x + \paren {\frac {6 x} {a^3} - \frac {x^3} a} \cos a x + C$
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:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \sin a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds -\frac {\cos a x} a\) | Primitive of $\sin a x$ |
Then:
\(\ds \int x^3 \sin a x \rd x\) | \(=\) | \(\ds x^3 \paren {-\frac {\cos a x} a} - \int 3 x^2 \paren {-\frac {\cos a x} a} \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac {x^3} a \cos a x + \frac 3 a \int x^2 \cos a x \rd x + C\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac {x^3} a \cos a x + \frac 3 a \paren {\frac {2 x} {a^2} \cos a x + \paren {\frac {x^2} a - \frac 2 {a^3} } \sin a x} + C\) | Primitive of $x^2 \cos a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \sin a x + \paren {\frac {6 x} {a^3} - \frac {x^3} a} \cos a x + C\) | simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$: $14.342$