Primitive of Exponential of a x by Sine of b x/Proof 2
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Theorem
- $\ds \int e^{a x} \sin b x \rd x = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C$
Proof
| \(\ds \cos b x + i \sin b x\) | \(=\) | \(\ds e^{i b x}\) | Euler's Formula | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds e^{a x} \cos b x + i e^{a x} \sin b x\) | \(=\) | \(\ds e^{a x} e^{i b x}\) | multiplying both sides by $e^{a x}$ | ||||||||||
| \(\ds \) | \(=\) | \(\ds e^{\paren {a + i b} x}\) | Exponent Combination Laws | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int e^{a x} \cos b x \, \d x + i \int e^{a x} \sin b x \d x\) | \(=\) | \(\ds \int e^{\paren {a + i b} x} \d x\) | Linear Combination of Complex Integrals | ||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {a + i b} e^{\paren {a + i b} x} + C\) | Primitive of $e^{a x}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {a - i b} {a^2 + b^2} e^{\paren {a + i b} x} + C\) | multiplying top and bottom by $a - i b$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {a - i b} {a^2 + b^2} e^{a x} e^{i b x} + C\) | Exponent Combination Laws | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {a - i b} {a^2 + b^2} e^{a x} \paren {\cos b x + i \sin b x} + C\) | Euler's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac a {a^2 + b^2} e^{a x} \cos b x - \frac {i b} {a^2 + b^2} e^{a x} \cos b x\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac {i a} {a^2 + b^2} e^{a x} \sin b x + \frac b {a^2 + b^2} e^{a x} \sin b x + C\) |
The result follows from equating imaginary parts.
$\blacksquare$