Let $0<a<1$. Evaluate the integral
$$\int_0^\infty \frac{x^{a-1}}{1+x} dx.$$
2026-04-02 21:45:18.1775166318
Improper integral (using methods in complex variables)
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All integrals of the form $\displaystyle\int_0^\infty\frac{x^{a-1}}{1+x^b} dx~$ can be evaluated by substituting $t=\dfrac1{1+x^b}~,~$
then recognizing the expression of the beta function in the new integral, and employing
Euler's reflection formula for the $\Gamma$ function to rewrite the final result as $\dfrac\pi b~\csc\bigg(a~\dfrac\pi b\bigg)$.