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The systematic way to do these problems is to start by making the substitution . Then , so that . Also, as goes from 0 to , z goes round the unit circle, so that the integral becomes a contour integral round the unit circle. Finally, and .
For the first problem, , and so
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The only pole of the integrand comes at the origin, and the residue is obviously 6 (being the coefficient of ), so you should find it easy to use the residue theorem to get the value of the integral.
Do the second problem in the same way. You should find that
(after a bit of simplification). There are two poles inside the unit circle, at . Find the residue at each of them, and use the residue theorem to get the value of the integral.