How can the integral of x^p exp(-tx) be solved for p=3/4, 5/4, and 7/4?

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Homework Statement



Integral_0^1 x^p exp(-tx)dx where p=3/4. 5/4 and 7/4

Homework Equations



Can somebody tell me the final solution and the detailed procedure to get the final solution for this.

The Attempt at a Solution



I tried using gamma distribution and integration by parts but nothing worked.


Thanks
Kannan
 
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This can be expressed in terms of the incomplete gamma function (see wikipedia), but otherwise cannot be done in closed form.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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