# Primitive of Exponential of a x by Sine of b x/Proof 2

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## Theorem

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

## Proof

\(\displaystyle \cos b x + i \sin bx\) | \(=\) | \(\displaystyle e^{i b x}\) | Euler's Formula | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle e^{a x} \cos b x + i e^{a x} \sin b x\) | \(=\) | \(\displaystyle e^{a x} e^{i b x}\) | multiplying both sides by $e^{a x}$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle e^{\left({a + i b}\right)x}\) | Exponent Combination Laws | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \int e^{a x} \cos b x \, \mathrm d x + i \int e^{a x} \sin b x \rd x\) | \(=\) | \(\displaystyle \int e^{\left({a + i b}\right) x} \rd x\) | Linear Combination of Complex Integrals | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 {a + i b} e^{\paren {a + i b} x} + C\) | Primitive of $e^{a x}$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {a - i b} {a^2 + b^2} e^{\paren {a + i b} x} + C\) | multiplying top and bottom by $a - i b$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {a - i b} {a^2 + b^2} e^{a x} e^{i b x} + C\) | Exponent Combination Laws | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {a - i b} {a^2 + b^2} e^{a x} \paren {\cos b x + i \sin b x} + C\) | Euler's Formula | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac a {a^2 + b^2} e^{a x} \cos b x - \frac {i b} {a^2 + b^2} e^{a x} \cos b x\) | |||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \frac {i a} {a^2 + b^2} e^{a x} \sin b x + \frac b {a^2 + b^2} e^{a x} \sin b x + C\) |

The result follows from equating imaginary parts.

$\blacksquare$