Question

Integrate the function:

`sqrt(1-4x - x^2)`

Answer 1

Let `I = sqrt(1 - 4x - x^2)`

`= int sqrt(1 - (x^2 +  4x + 4) + 4)  dx`

`= int sqrt(5 - (x + 2)^2)  dx`

`= int sqrt ((5)^2 - (x+2)^2) dx`

Now, `[because  int sqrt (a^2 - x^2)dx = x/2 sqrt (a^2 - x^2) + a^2/2 sin^-1  x/a+C]`

Here, on putting 5 in place of a² and (x + 2) in place of x,

I = `1/2 (x + 2) sqrt(5 + (x + 2)^2) + 5/2  sin^-1  (x + 2)/sqrt5 + C`

`= 1/2 (x + 2) sqrt(1 - 4x - x^2) + 5/2  sin^-1  (x + 2)/sqrt5  + C`