If ddxG(x) = e^ xx, x (0,/2), then _1/4^1/2 2xe^( x^2),dx is equal to:

Question

If $\dfrac{d}{dx}G(x) = \dfrac{e^{	an x}}{x}$, $x \in (0,\pi/2)$, then $\displaystyle\int_{1/4}^{1/2} \dfrac{2}{x}e^{	an(\pi x^2)}\,dx$ is equal to:

Accepted Answer

$I = \int_{1/4}^{1/2} \dfrac{2}{x}e^{	an(\pi x^2)}\,dx = \int_{1/4}^{1/2} \dfrac{2\pi x}{\pi x^2}e^{	an(\pi x^2)}\,dx$

Let $\pi x^2=t \Rightarrow 2\pi x\,dx=dt$; when $x=\dfrac12$, $t=\dfrac{\pi}{4}$; when $x=\dfrac14$, $t=\dfrac{\pi}{16}$

$I = \int_{\pi/16}^{\pi/4} \dfrac{e^{	an t}}{t}\,dt = G(t)\Big|_{\pi/16}^{\pi/4} = G\left(\dfrac{\pi}{4}ight)-G\left(\dfrac{\pi}{16}ight)$

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