Primitive of Exponential of a x by Power of Cosine of b x
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Theorem
- $\ds \int e^{a x} \cos^n b x \rd x = \frac {e^{a x} \cos^{n - 1} b x} {a^2 + n^2 b^2} \paren {a \cos b x + n b \sin b x} + \frac {n \paren {n - 1} b^2} {a^2 + n^2 b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C$
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:
\(\ds u\) | \(=\) | \(\ds \cos^n b x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds -n b \cos^{n - 1} b x \sin b x\) | Derivative of $\cos a x$, Derivative of Power, Chain Rule for Derivatives |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds e^{a x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {e^{a x} } a\) | Primitive of $e^{a x}$ |
Then:
\(\ds \int e^{a x} \cos^n b x \rd x\) | \(=\) | \(\ds \cos^n b x \paren {\frac {e^{a x} } a} - \int \paren {\frac {e^{a x} } a} \paren {-n b \cos^{n - 1} b x \sin b x} \rd x + C\) | Integration by Parts | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \frac {e^{a x} \cos^n b x} a + \frac {n b} a \int e^{a x} \cos^{n - 1} b x \sin b x \rd x + C\) | Primitive of Constant Multiple of Function |
From Primitive of $e^{a x} \cos^{n - 1} b x \sin b x$:
- $\ds \int e^{a x} \cos^{n - 1} b x \sin b x \rd x = \frac {e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x} } {a^2 + n b^2} - \frac {\paren {n - 1} a b} {a^2 + n b^2} \paren {\int e^{a x} \cos^n b x \rd x - \int e^{a x} \cos^{n - 2} b x \rd x} + C$
Hence:
\(\ds \) | \(\) | \(\ds \int e^{a x} \cos^n b x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} \cos^n b x} a + \frac {n b} a \int e^{a x} \cos^{n - 1} b x \sin b x \rd x + C\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} \cos^n b x} a + \frac {n b} {a \paren {a^2 + n b^2} } e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x}\) | Primitive of $e^{a x} \cos^n b x$: Lemma 1 | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac {n \paren {n - 1} b^2} {a^2 + n b^2} \int e^{a x} \cos^n b x \rd x + \frac {n \paren {n - 1} b^2} {a^2 + n b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds \paren {1 + \frac {n \paren {n - 1} b^2} {a^2 + n b^2} } \int e^{a x} \cos^n b x \rd x\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} \cos^n b x} a + \frac {n b} {a \paren {a^2 + n b^2} } e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac {n \paren {n - 1} b^2} {a^2 + n b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds \frac {a^2 + n b^2 + n^2 b^2 - n b^2} {a^2 + n b^2} \int e^{a x} \cos^n b x \rd x\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} \cos^n b x} a + \frac {n b} {a \paren {a^2 + n b^2} } e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac {n \paren {n - 1} b^2} {a^2 + n b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds \paren {a^2 + n^2 b^2} \int e^{a x} \cos^n b x \rd x\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {a^2 + n b^2} a e^{a x} \cos^n b x + \frac {n b} a e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {n \paren {n - 1} b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C\) | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{a x} \cos^{n - 1} b x \paren {a \cos b x + n b \sin b x}\) | Primitive of $e^{a x} \cos^n b x$: Lemma 2 | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {n \paren {n - 1} b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds \int e^{a x} \cos^n b x \rd x\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} \cos^{n - 1} b x} {a^2 + n^2 b^2} \paren {a \cos b x + n b \sin b x} + \frac {n \paren {n - 1} b^2} {a^2 + n^2 b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C\) |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $e^{a x}$: $14.524$