Primitive of Exponential of a x by Sine of b x

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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 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
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C\)

$\blacksquare$


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