Primitive of Exponential of a x by Power of Cosine of b x

From ProofWiki
Jump to navigation Jump to search

Theorem

$\displaystyle \int e^{a x} \cos^n b x \rd x = \frac {e^{a x} \cos^{n - 1} b x} {a^2 + n^2 b^2} \paren {a \cos b x + n b \sin b x} + \frac {n \paren {n - 1} b^2} {a^2 + n^2 b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C$


Proof

With a view to expressing the primitive in the form:

$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

\(\displaystyle u\) \(=\) \(\displaystyle \cos^n b x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d u} {\d x}\) \(=\) \(\displaystyle -n b \cos^{n - 1} b x \sin b x\) Derivative of $\cos a x$, Derivative of Power, Chain Rule for Derivatives


and let:

\(\displaystyle \frac {\mathrm d v} {\mathrm d x}\) \(=\) \(\displaystyle e^{a x}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle \frac {e^{a x} } a\) Primitive of $e^{a x}$


Then:

\(\displaystyle \int e^{a x} \cos^n b x \rd x\) \(=\) \(\displaystyle \cos^n b x \paren {\frac {e^{a x} } a} - \int \paren {\frac {e^{a x} } a} \paren {-n b \cos^{n - 1} b x \sin b x} \rd x + C\) Integration by Parts
\(\text {(1)}: \quad\) \(\displaystyle \) \(=\) \(\displaystyle \frac {e^{a x} \cos^n b x} a + \frac {n b} a \int e^{a x} \cos^{n - 1} b x \sin b x \rd x + C\) Primitive of Constant Multiple of Function


From Primitive of $e^{a x} \cos^{n - 1} b x \sin b x$:

$\displaystyle \int e^{a x} \cos^{n - 1} b x \sin b x \rd x = \frac {e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x} } {a^2 + n b^2} - \frac {\paren {n - 1} a b} {a^2 + n b^2} \paren {\int e^{a x} \cos^n b x \rd x - \int e^{a x} \cos^{n - 2} b x \rd x} + C$


Hence:

\(\displaystyle \) \(\) \(\displaystyle \int e^{a x} \cos^n b x \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{a x} \cos^n b x} a + \frac {n b} a \int e^{a x} \cos^{n - 1} b x \sin b x \rd x + C\) from $(1)$
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{a x} \cos^n b x} a + \frac {n b} {a \paren {a^2 + n b^2} } e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x}\) Primitive of $e^{a x} \cos^n b x$: Lemma 1
\(\displaystyle \) \(\) \(\, \displaystyle - \, \) \(\displaystyle \frac {n \paren {n - 1} b^2} {a^2 + n b^2} \int e^{a x} \cos^n b x \rd x + \frac {n \paren {n - 1} b^2} {a^2 + n b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \) \(\) \(\displaystyle \paren {1 + \frac {n \paren {n - 1} b^2} {a^2 + n b^2} } \int e^{a x} \cos^n b x \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{a x} \cos^n b x} a + \frac {n b} {a \paren {a^2 + n b^2} } e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac {n \paren {n - 1} b^2} {a^2 + n b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \) \(\) \(\displaystyle \frac {a^2 + n b^2 + n^2 b^2 - n b^2} {a^2 + n b^2} \int e^{a x} \cos^n b x \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{a x} \cos^n b x} a + \frac {n b} {a \paren {a^2 + n b^2} } e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac {n \paren {n - 1} b^2} {a^2 + n b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \) \(\) \(\displaystyle \paren {a^2 + n^2 b^2} \int e^{a x} \cos^n b x \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2 + n b^2} a e^{a x} \cos^n b x + \frac {n b} a e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \paren {n \paren {n - 1} b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \) \(\) \(\displaystyle \paren {a^2 + n^2 b^2} \int e^{a x} \cos^n b x \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle e^{a x} \cos^{n - 1} b x \paren {a \cos b x + n b \sin b x}\) Primitive of $e^{a x} \cos^n b x$: Lemma 2
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \paren {n \paren {n - 1} b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \) \(\) \(\displaystyle \int e^{a x} \cos^n b x \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{a x} \cos^{n - 1} b x} {a^2 + n^2 b^2} \paren {a \cos b x + n b \sin b x} + \frac {n \paren {n - 1} b^2} {a^2 + n^2 b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C\)

$\blacksquare$


Also see


Sources