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Question
Integrate the function:
`sqrt(1-4x - x^2)`
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Solution
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`
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