Decide whether each integral converges. If the integral converges, compute its value.
Choose the correct letter for each question.

-converges to $\frac { 1 } { 4 }$
A)$\int _ { 0 } ^ { + \infty } \sin ( 2 x ) \mathrm { d } x$
B)$\int _ { 0 } ^ { + \infty } e ^ { - 2 x } \sin ( 2 x ) \mathrm { d } x$

Question

Decide whether each integral converges. If the integral converges, compute its value.
Choose the correct letter for each question.

-converges to $\frac { 1 } { 4 }$
A)$\int _ { 0 } ^ { + \infty } \sin ( 2 x ) \mathrm { d } x$
B)$\int _ { 0 } ^ { + \infty } e ^ { - 2 x } \sin ( 2 x ) \mathrm { d } x$

Quizplus · Accepted Answer

The integral in option B, $\int _ { 0 } ^ { + \infty } e ^ { - 2 x } \sin ( 2 x ) \mathrm { d } x$, converges to $\frac{1}{4}$ because it is an example of an exponential decay function multiplied by a sine function, which is a common form of an integral that converges to a finite value. The presence of the exponential term $e^{-2x}$ ensures that the integral converges as $x$ approaches infinity, unlike the integral in option A, which does not converge due to the sine function oscillating indefinitely without a decaying factor.

Decide Whether Each Integral Converges 14\frac { 1 } { 4 }41​

Decide Whether Each Integral Converges $\frac { 1 } { 4 }$