Question

Evaluate the integral with hyperbolic or trigonometric substitution. ?

Answer 1

`-sqrt(4-x^2)/x-sin^-1(x/2)+C`

Explanation:

Substitute `x = 2 sin t`. Then `sqrt(4-x^2) = 2 cos t` and `dx = 2 cos t dt`

Thus

`int sqrt(4-x^2)/x^2 = int (2cos t)/(4sin^2t)2cos t dt = int cot^2 t dt`
`= int (csc^2t-1)dt = -cot t - t+C`
`= -sqrt(4-x^2)/x-sin^-1(x/2)+C`