Primitive of Exponential of a x by Power of Sine of b x/Lemma 1
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Lemma for Primitive of $e^{a x} \sin^n b x \cos b x$
- $\ds \int e^{a x} \sin^{n - 1} b x \cos b x \rd x = \frac {e^{a x} \sin^{n - 1} b x \paren {a \cos b x + b \sin b x} } {a^2 + n b^2} + \frac {\paren {n - 1} a b} {a^2 + n b^2} \paren {\int e^{a x} \sin^n b x \rd x - \int e^{a x} \sin^{n - 2} b x \rd x} + C$
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:
\(\ds u\) | \(=\) | \(\ds \sin^{n - 1} b x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \paren {n - 1} b \sin^{n - 2} b x \cos b x\) | Derivative of $\sin a x$, Derivative of Power, Chain Rule for Derivatives |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds e^{a x} \cos b x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2}\) | Primitive of $e^{a x} \cos b x$ |
Then:
\(\ds \) | \(\) | \(\ds \int e^{a x} \sin^{n - 1} b x \cos b x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {\sin^{n - 1} b x} \paren {e^{a x} \cos b x} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sin^{n - 1} b x} \paren {\frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} }\) | Integration by Parts | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \int \paren {\frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} } \paren {\paren {n - 1} b \sin^{n - 2} b x \cos b x} \rd x + C\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} \sin^{n - 1} b x \paren {a \cos b x + b \sin b x} } {a^2 + b^2}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac {\paren {n - 1} a b} {a^2 + b^2} \int e^{a x} \sin^{n - 2} b x \cos^2 b x \rd x + C\) | Linear Combination of Primitives | ||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac {\paren {n - 1} b^2} {a^2 + b^2} \int e^{a x} \sin^{n - 1} b x \cos b x \rd x + C\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds \paren {1 + \frac {\paren {n - 1} b^2} {a^2 + b^2} } \int e^{a x} \sin^{n - 1} b x \cos b x \rd x\) | gathering terms | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} \sin^{n - 1} b x \paren {a \cos b x + b \sin b x} } {a^2 + b^2}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac {\paren {n - 1} a b} {a^2 + b^2} \int e^{a x} \sin^{n - 2} b x \cos^2 b x \rd x + C\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds \paren {a^2 + n b^2} \int e^{a x} \sin^{n - 1} b x \cos b x \rd x\) | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{a x} \sin^{n - 1} b x \paren {a \cos b x + b \sin b x}\) | simplifying | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren {n - 1} a b \int e^{a x} \sin^{n - 2} b x \cos^2 b x \rd x + C\) | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{a x} \sin^{n - 1} b x \paren {a \cos b x + b \sin b x}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren {n - 1} a b \int e^{a x} \sin^{n - 2} b x \paren {1 - \sin^2 b x} \rd x + C\) | Sum of Squares of Sine and Cosine | ||||||||||
\(\ds \) | \(=\) | \(\ds e^{a x} \sin^{n - 1} b x \paren {a \cos b x + b \sin b x}\) | simplifying | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren {n - 1} a b \int e^{a x} \sin^{n - 2} b x \rd x\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {n - 1} a b \int e^{a x} \sin^n b x \rd x + C\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds \int e^{a x} \sin^{n - 1} b x \cos b x \rd x\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} \sin^{n - 1} b x \paren {a \cos b x + b \sin b x} } {a^2 + n b^2}\) | simplifying | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac {\paren {n - 1} a b} {a^2 + n b^2} \int e^{a x} \sin^{n - 2} b x \rd x\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac {\paren {n - 1} a b} {a^2 + n b^2} \int e^{a x} \sin^n b x \rd x + C\) |
and so rearranging:
- $\ds \int e^{a x} \sin^{n - 1} b x \cos b x \rd x = \frac {e^{a x} \sin^{n - 1} b x \paren {a \cos b x + b \sin b x} } {a^2 + n b^2} + \frac {\paren {n - 1} a b} {a^2 + n b^2} \paren {\int e^{a x} \sin^n b x \rd x - \int e^{a x} \sin^{n - 2} b x \rd x} + C$
$\blacksquare$