Extension of definite integrals

Exercise

1.

Evaluate the following improper integrals．

(a)

$${\int_{0}^{1}\frac{dx}{\sqrt{1-x}}}$$

(b)

$${\int_{0}^{1}\frac{dx}{x}}$$

(c)

$${\int_{0}^{1}\log{x}dx}$$

2.

Evaluate the following improper integrals．

(a)

$${\int_{0}^{\infty}xe^{-x}dx}$$

(b)

$${\int_{2}^{\infty}\frac{1}{x(\log{x})^{\alpha}}dx}$$

(c)

$${\int_{2}^{\infty}\frac{1}{x^{\alpha} \log{x}} dx}$$

3.

Evaluate the convergence or divergence of the following integrals．

(a)

$${\int_{0}^{\frac{\pi}{2}}\frac{dx}{\sqrt{\cos{x}}}}$$

(b)

$${\int_{0}^{1}\frac{\log{x}}{\sqrt{x}}}$$