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

Question

Quizplus · Accepted Answer

Integration by parts for $\int e^{ax} \sin bx dx$ requires applying the formula twice due to the cyclic nature of integrating $e^{ax}$ and $\sin bx$ or $\cos bx$. The correct result involves a combination of $e^{ax}$, $\sin bx$, and $\cos bx$ with coefficients that ensure the integral's differentiation leads back to the original function. The correct answer, therefore, involves both sine and cosine terms with coefficients related to $a$ and $b$, making option D the correct choice as it reflects the structure and coefficients resulting from the integration by parts process.

Evaluate the Integral Using Integration by Parts ∫eaxsin⁡bxdx\int e ^ { a x } \sin b x d x∫eaxsinbxdx

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