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the croods 2013 greek audio downloadQ:

Evaluate the definite integral

Evaluate the integral $$\int_0^1\sqrt1-te^ -\frac1tdt$$

I tried integrating by parts, but I didn't really get anywhere.

I then tried evaluating it using the gamma function, but this fails to give me an answer

A:

$$ \int_0^1\sqrt1-te^ -\frac1tdt= \int_0^1\sqrt1-te^ -\frac11-tdt = \int_0^1 \sqrt\frac1-t1-te^\frac11-tdt=\int_0^1 \frac1-t1-t e^\frac11-tdt =\int_0^1 \frac1t e^\frac1tdt=\left[-e^\frac1t\right]_0^1 =-\frac1e $$

A:

Another approach is to use the substitution $s=1-t$:

$$\beginalign*

\int_0^1\sqrt1-t\,e^ -\frac1t\,\mathrmdt&=\int_0^1(1-s)\sqrt1-s\,e^\frac1s\,\mathrmds\\

&=\int_0^1\sqrts\,e^\frac1s\,\mathrmds

\endalign*$$

where we used the substitution $s=1-t$ to obtain the second line.

On the second line we can use the Gamma function since $s=1-t$ so that $\mathrm{

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