Solve the problem.
-Evaluate
$$\int _ { 0 } ^ { \infty } \frac { \sin a x } { x } d x , a 0$$
by integrating
$$\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } e ^ { - x y } \sin a x d A$$
A) $\frac { \pi a } { 4 }$
B) $\frac { \pi a } { 6 }$
C) $\frac { \pi } { 2 }$
D) $\frac { \pi \mathrm { a } } { 3 }$

Question

Solve the problem.
-Evaluate
$$\int _ { 0 } ^ { \infty } \frac { \sin a x } { x } d x , a > 0$$
by integrating
$$\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } e ^ { - x y } \sin a x d A$$
A) $\frac { \pi a } { 4 }$
B) $\frac { \pi a } { 6 }$
C) $\frac { \pi } { 2 }$
D) $\frac { \pi \mathrm { a } } { 3 }$

Quizplus · Accepted Answer

First, we integrate the given double integral:
\begin{align*} \int_0^\infty \int_0^\infty e^{-xy} \sin ax~dxdy &= \int_0^\infty \int_0^\infty e^{-xy} \sin ax~dydx\ &= \int_0^\infty \left(\int_0^\infty e^{-xy} \sin ax~dx\right)dy\ &= \int_0^\infty \frac{a}{a^2+y^2}~dy\ &= \frac{\pi}{2} \end{align*}
by using the substitution $u=ax$. On the other hand, we can write the original integral as
$$\int_0^\infty \frac{\sin ax}{x}~dx = \int_0^\infty \int_0^\infty e^{-xy} \sin ax~dxdy$$
Therefore, combining the above results gives us
$$\int_0^\infty \frac{\sin ax}{x}~dx = \frac{\pi}{2}$$
which is choice C.

Solve the Problem ∫0∞sin⁡axxdx,a>0\int _ { 0 } ^ { \infty } \frac { \sin a x } { x } d x , a > 0∫0∞​xsinax​dx,a>0

Solve the Problem $\int _ { 0 } ^ { \infty } \frac { \sin a x } { x } d x , a > 0$