Extension of definite integrals

Exercise

1.: Evaluate the following improper integrals．
(a): $\displaystyle{\int_{0}^{1}\frac{dx}{\sqrt{1-x}}}$
(b): $\displaystyle{\int_{0}^{1}\frac{dx}{x}}$
(c): $\displaystyle{\int_{0}^{1}\log{x}dx}$
2.: Evaluate the following improper integrals．
(a): $\displaystyle{\int_{0}^{\infty}xe^{-x}dx}$
(b): $\displaystyle{\int_{2}^{\infty}\frac{1}{x(\log{x})^{\alpha}}dx}$
(c): $\displaystyle{\int_{2}^{\infty}\frac{1}{x^{\alpha} \log{x}} dx}$
3.: Evaluate the convergence or divergence of the following integrals．
(a): $\displaystyle{\int_{0}^{\frac{\pi}{2}}\frac{dx}{\sqrt{\cos{x}}}}$
(b): $\displaystyle{\int_{0}^{1}\frac{\log{x}}{\sqrt{x}}}$