The improper integral $\int _ { 0 } ^ { \infty } e ^ { - x } \sin x d x$ is

A) $- \infty$
B) $\frac { \pi } { 4 }$
C) $\frac { 1 } { 2 }$
D) $\pi$
E) $\infty$

Question

The improper integral $\int _ { 0 } ^ { \infty } e ^ { - x } \sin x d x$ is

A) $- \infty$
B) $\frac { \pi } { 4 }$
C) $\frac { 1 } { 2 }$
D) $\pi$
E) $\infty$

Quizplus · Accepted Answer

This integral can be solved using integration by parts or recognizing it as a standard form integral. The integral of $e^{-x}\sin(x)dx$ from 0 to $\infty$ converges to $\frac{1}{2}$, as it represents the imaginary part of the integral of $e^{-(1-i)x}$ from 0 to $\infty$, which equals $\frac{1}{1+i} = \frac{1}{2} + \frac{1}{2}i$.

The Improper Integral ∫0∞e−xsin⁡xdx\int _ { 0 } ^ { \infty } e ^ { - x } \sin x d x∫0∞​e−xsinxdx

The Improper Integral $\int _ { 0 } ^ { \infty } e ^ { - x } \sin x d x$