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## Homework Statement

If the value of the integral ##\displaystyle \int_1^2 e^{x^2}\,\, dx## is ##\alpha##, then the value of ##\displaystyle \int_e^{e^4} \sqrt{\ln x} \,\, dx## is:

A)##e^4-e-\alpha##

B)##2e^4-e-\alpha##

C)##2(e^4-e)-\alpha##

D)##2e^4-1-\alpha##

## Homework Equations

## The Attempt at a Solution

Starting with the given integral, I used the substitution, ##e^{x^2}=t\Rightarrow 2xe^{x^2}dx=dt##.

$$\int_1^2 e^{x^2} dx=\int_1^2 \frac{2xe^{x^2}}{2x}dx=\frac{1}{2}\int_e^{e^4} \frac{dt}{\sqrt{\ln t}}$$

But this doesn't end up with the definite integral asked in the problem. :(

I have tried using the substitution ##\sqrt{\ln x}=t## in the definite integral to be evaluated, I end up with ##\displaystyle \int_1^2 t^2\cdot e^{t^2} dt## but this isn't the same as given in the problem statement.

Any help is appreciated. Thanks!