# Primitive of Exponential of a x by Sine of b x

## 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 1

 $\displaystyle \int e^{a x} \sin b x \rd x$ $=$ $\displaystyle \frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x$ Primitive of $e^{a x} \sin b x$: Lemma $\displaystyle$ $=$ $\displaystyle \frac {e^{a x} \sin b x} a - \frac b a \paren {\frac {e^{a x} \cos b x} a + \frac b a \int e^{a x} \sin b x \rd x}$ Primitive of $e^{a x} \cos b x$: Lemma $\displaystyle$ $=$ $\displaystyle \frac {e^{a x} a \sin b x - e^{a x} b \cos b x} {a^2} - \frac {b^2} {a^2} \int e^{a x} \sin b x \rd x$ simplifying $\displaystyle \leadsto \ \$ $\displaystyle \paren {1 + \frac {b^2} {a^2} } \int e^{a x} \sin b x \rd x$ $=$ $\displaystyle \frac {e^{a x} \left({a \sin b x - b \cos b x}\right)} {a^2}$ simplifying $\displaystyle \leadsto \ \$ $\displaystyle \frac {a^2 + b^2} {a^2} \int e^{a x} \sin b x \rd x$ $=$ $\displaystyle \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2}$ common denominator $\displaystyle \leadsto \ \$ $\displaystyle \int e^{a x} \sin b x \rd x$ $=$ $\displaystyle \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2}$ multiplying by $\dfrac {a^2} {a^2 + b^2}$

$\blacksquare$

## Proof 2

 $\displaystyle \cos b x + i \sin bx$ $=$ $\displaystyle e^{i b x}$ 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.

$\blacksquare$

## Proof 3

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

$\blacksquare$