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.

$\blacksquare$