Integration of irrational functions

Exercise

1.

Find the following integrals．

(a)

$${\int{x \sqrt{1 + x}}\ dx}$$

(b)

$${\int{\frac{\sqrt{x}}{\sqrt{x} - 1}}\ dx}$$

(c)

$${\int{\frac{dx}{\sqrt{1 + e^x}}}}$$

(d)

$${\int{x^2 \sqrt{x - 1}}\ dx}$$

(e)

$${\int{\sqrt{\frac{x+1}{x-1}}}\ dx}$$

2.

Find the following integrals．

(a)

$${\int{\frac{x}{\sqrt{x^2 - 4}}}\ dx}$$

(b)

$${\int{\frac{x^2}{\sqrt{4 - x^2}}}\ dx}$$

(c)

$${\int{\frac{e^x}{9 - e^{2x}}}\ dx}$$

(d)

$${\int{\frac{\sqrt{1 - x^2}}{x^4}}\ dx}$$

(e)

$${\int{\frac{dx}{x^2\sqrt{x^2 - a^2}}}}$$

(f)

$${\int{\frac{dx}{e^x\sqrt{4 + e^{2x}}}}}$$

(g)

$${\int{\frac{dx}{\sqrt{x^2 - 2x - 3}}}}$$

(h)

$${\int{\frac{x}{\sqrt{6x - x^2}}}\ dx}$$

(i)

$${\int{\frac{x}{\sqrt{x^2 - 2x - 3}}}\ dx}$$

(j)

$${\int{\sqrt{6x - x^2 - 8}}\ dx}$$

(k)

$${\int{x\sqrt{x^2 + 6x}}\ dx}$$