Primitive of Exponential of a x by Cosine of b x/Proof 1
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Theorem
- $\ds \int e^{a x} \cos b x \rd x = \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} + C$
Proof
\(\ds \int e^{a x} \cos b x \rd x\) | \(=\) | \(\ds \frac {e^{a x} \cos b x} a + \frac b a \int e^{a x} \sin b x \rd x\) | Primitive of $e^{a x} \cos b x$: Lemma | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} \cos b x} a + \frac b a \paren {\frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x}\) | Primitive of $e^{a x} \sin b x$: Lemma | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} a \cos b x + e^{a x} b \sin b x} {a^2} - \frac {b^2} {a^2} \int e^{a x} \cos b x \rd x\) | simplifying | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {1 + \frac {b^2} {a^2} } \int e^{a x} \cos b x \rd x\) | \(=\) | \(\ds \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2}\) | simplifying | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {a^2 + b^2} {a^2} \int e^{a x} \cos b x \rd x\) | \(=\) | \(\ds \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2}\) | common denominator | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int e^{a x} \cos b x \rd x\) | \(=\) | \(\ds \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2}\) | multiplying by $\dfrac {a^2} {a^2 + b^2}$ |
$\blacksquare$