Primitive of x by Exponential of a x by Sine of b x
Jump to navigation
Jump to search
Theorem
- $\ds \int x e^{a x} \sin b x \rd x = \frac {x e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} - \frac {e^{a x} \paren {\paren {a^2 - b^2} \sin b x - 2 a b \cos b x} } {\paren {a^2 + b^2}^2} + C$
Proof
With a view to expressing the primitive in the form:
- $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds 1\) | Derivative of Identity Function |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds e^{a x} \sin b x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2}\) | Primitive of $e^{a x} \sin b x$ |
Then:
\(\ds \int x e^{a x} \sin b x \rd x\) | \(=\) | \(\ds x \paren {\frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} } - \int \paren {\frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} } \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac a {a^2 + b^2} \int e^{a x} \sin b x \rd x + \frac b {a^2 + b^2} \int e^{a x} \cos b x \rd x + C\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac a {a^2 + b^2} \paren {\frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} }\) | Primitive of $e^{a x} \sin b x$ | ||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac b {a^2 + b^2} \paren {\frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} } + C\) | Primitive of $e^{a x} \cos b x$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} - \frac {e^{a x} \paren {\paren {a^2 - b^2} \sin b x - 2 a b \cos b x} } {\paren {a^2 + b^2}^2} + C\) | simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $e^{a x}$: $14.520$