Primitive of Exponential of a x by Cosine of b x/Proof 2
Jump to navigation
Jump to search
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
\(\ds \int e^{a x} e^{i b x} \rd x\) | \(=\) | \(\ds i \int e^{a x} \sin b x \rd x + \int e^{a x} \cos b x \rd x\) | Euler's Formula | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int e^{a x} \cos b x \rd x\) | \(=\) | \(\ds \map \Re {\int e^{\paren {a + i b} x} \rd x}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {\frac {e^{\paren {a + i b} x} } {a + i b} } + C\) | Primitive of Exponential of a x | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {\frac {\paren {a - i b} e^{\paren {a + i b} x} } {a^2 + b^2} } + C\) | multiplying through by $\dfrac {a - i b} {a - i b}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {\frac {i a e^{a x} \sin b x + a e^{a x} \cos b x - i b \paren {i e^{a x} \sin b x + e^{a x} \cos b x} } {a^2 + b^2} } + C\) | Euler's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {\frac {i \paren {a e^{a x} \sin b x - b e^{a x} \cos b x} + \paren {a e^{a x} \cos b x + b e^{a x} \sin b x} } { a^2 + b^2} } + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} + C\) | isolating real part |
$\blacksquare$