Primitive of Power of x by Sine of a x/Lemma

Theorem

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

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:

\(\ds u\)

\(=\)

\(\ds x^m\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\d u} {\d x}\)

\(=\)

\(\ds m x^{m - 1}\)

Power Rule for Derivatives

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:

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

\(=\)

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

Integration by Parts

\(\ds \)

\(=\)

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

Primitive of Constant Multiple of Function

$\blacksquare$

Also see

• Primitive of $x^m \cos a x$

Sources

• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $80$.