Primitive of Power of x by Sine of a x

Theorem

$\ds \int x^m \sin a x \rd x = \frac {-x^m \cos a x} a + \frac {m x^{m - 1} \sin a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \int x^{m - 2} \sin a x \rd x$

Proof

Lemma

$\ds \int x^m \sin a x \rd x = \frac {- x^m \cos a x} a + \frac m a \int x^{m - 1} \cos a x \rd x$

$\Box$

From Primitive of $x^{m - 1} \cos a x$: Lemma:

$(1): \quad \ds \int x^{m - 1} \cos a x \rd x = \frac {x^{m - 1} \sin a x} a - \frac {m - 1} a \int x^{m - 2} \sin a x \rd x$

So:

\(\ds \int x^m \sin a x \rd x\)

\(=\)

\(\ds \frac {- x^m \cos a x} a + \frac m a \int x^{m - 1} \cos a x \rd x\)

Lemma

\(\ds \)

\(=\)

\(\ds \frac {- x^m \cos a x} a + \frac m a \paren {\frac {x^{m - 1} \sin a x} a - \frac {m - 1} a \int x^{m - 2} \sin a x \rd x}\)

from $(1)$

\(\ds \)

\(=\)

\(\ds \frac {- x^m \cos a x} a + \frac {m x^{m - 1} \sin a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \int x^{m - 2} \sin a x \rd x\)

simplifying

$\blacksquare$

Also see

• Primitive of $x^m \cos a x$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$: $14.364$