Evaluate the integral using integration by parts.
-$\int e ^ { a x } \cos b x d x$
A) $\frac { 1 } { 2 b }$ $\mathrm { e } ^ { \mathrm { ax } }$ $\cos ^ { 2 }$ bx + C
B) $\mathrm { e } ^ { \mathrm { ax } }$ (sin bx - cos bx) + C
C) $\mathrm { e } ^ { \mathrm { ax } }$ $\left( \frac { b \sin b x + a \cos b x } { a ^ { 2 } + b ^ { 2 } } \right)$ + C
D) $\mathrm { e } ^ { \mathrm { ax } }$ (sin bx + cos bx) + C

Question

Quizplus · Accepted Answer

Integration by parts for $\int e^{ax} \cos bx dx$ requires applying the formula twice, leading to a solution that combines both sine and cosine functions with coefficients related to $a$ and $b$, and is normalized by the sum of their squares, $a^2 + b^2$, as shown in option C.

Evaluate the Integral Using Integration by Parts ∫eaxcos⁡bxdx\int e ^ { a x } \cos b x d x∫eaxcosbxdx

Evaluate the Integral Using Integration by Parts $\int e ^ { a x } \cos b x d x$