Primitive of Exponential of a x by Sine of b x/Proof 3

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Theorem

$\displaystyle \int e^{a x} \sin b x \ \mathrm d x = \frac {e^{a x} \left({a \sin b x - b \cos bx}\right)} {a^2 + b^2} + C$


Proof

\(\displaystyle \int e^{a x} \sin b x \ \mathrm d x\) \(=\) \(\displaystyle \int e^{a x} \left({\frac {e^{i b x} - e^{-i b x} }{2 i} }\right) \ \mathrm d x\) $\quad$ Sine Exponential Formulation $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 i} \int e^{a x} \left({e^{i b x} - e^{-i b x} }\right) \ \mathrm d x\) $\quad$ Primitive of Constant Multiple of Function $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 i} \int \left({e^{a x} e^{i b x} - e^{a x} e^{-i b x} }\right) \ \mathrm d x\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 i} \int \left({e^{a x + i b x} - e^{a x - i b x} }\right) \ \mathrm d x\) $\quad$ Exponent Combination Laws: Product of Powers $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 i} \int e^{a x + i b x} \ \mathrm d x - \frac 1 {2 i} \int e^{a x - i b x} \ \mathrm d x\) $\quad$ Linear Combination of Integrals $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 i} \int e^{\left({a + i b}\right) x} \ \mathrm d x - \frac 1 {2 i} \int e^{\left({a - i b}\right) x} \ \mathrm d x\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 i} \frac {e^{\left({a + i b}\right) x} } {a + i b} - \frac 1 {2 i} \frac {e^{\left({a - i b}\right) x} } {a - i b} + C\) $\quad$ Primitive of $e^{a x}$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 i} \frac {e^a e^{i b} x} {a + i b} - \frac 1 {2 i} \frac {e^a e^{-i b} x} {a - i b} + C\) $\quad$ Exponent Combination Laws: Product of Powers $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 i} \frac {e^a e^{i b} x \left({a - i b}\right)} {\left({a + i b}\right) \left({a - i b}\right)} - \frac 1 {2 i} \frac {e^a e^{-i b} x \left({a + i b}\right)} {\left({a - i b}\right) \left({a + i b}\right)} + C\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^a e^{i b} x \left({a - i b}\right) - e^a e^{-i b} x \left({a + i b}\right)} {2 i \left({a + i b}\right) \left({a - i b}\right)} + C\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^a e^{i b} x \left({a - i b}\right) - e^a e^{-i b} x \left({a + i b}\right)} {2 i \left({a^2 + b^2}\right)} + C\) $\quad$ Product of Complex Number with Conjugate $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^a e^{i b} a x - e^a e^{i b} i b x - e^a e^{-i b} a x - e^a e^{-i b} i b x} {2 i \left({a^2 + b^2}\right)} + C\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^a} {\left({a^2 + b^2}\right)} \left({\frac {e^{i b} a x - e^{i b} i b x - e^{-i b} a x - e^{-i b} i b x} {2 i} }\right) + C\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^a} {\left({a^2 + b^2}\right)} \left({a \frac {e^{i b} x - e^{-i b} x} {2 i} - b \frac{e^{i b} x + e^{-i b} b x} 2 }\right) + C\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^a} {\left({a^2 + b^2}\right)} \left({a \frac {e^{i b} x - e^{-i b} x} {2 i} - b \cos b x }\right) + C\) $\quad$ Cosine Exponential Formulation $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^a} {\left({a^2 + b^2}\right)} \left({a \sin b x - b \cos b x }\right) + C\) $\quad$ Sine Exponential Formulation $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{a x} \left({a \sin b x - b \cos bx}\right)} {a^2 + b^2} + C\) $\quad$ $\quad$

$\blacksquare$