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

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

Question

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

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

Quizplus · Accepted Answer

The integral $\int _ { 0 } ^ { \infty } e ^ { - x } \cos x d x$ can be solved using integration by parts or by recognizing it as the real part of the integral of $e^{-x}e^{ix}$, which equals $\int _ { 0 } ^ { \infty } e ^ { - (1 - i) x } d x$. This evaluates to $\frac{1}{1-i}$, which simplifies to $\frac{1}{2} + \frac{i}{2}$. The real part of this result is $\frac{1}{2}$, making option C correct.

The Improper Integral ∫0∞e−xcos⁡xdx\int _ { 0 } ^ { \infty } e ^ { - x } \cos x d x∫0∞​e−xcosxdx

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