We may factor the denominator of the integrand:
Now the method of Partial Fractions may be used to split the integrand into the sum of two fractions:
the integral is Integral sign ("S") dx/ (sqrt(21-4x-x^2))
So first I completed the square so 21 - (x+2)^2 +4 so
Let u = x+2 so du = dx
so the original integral becomes S du/ sqrt( 25 - u^2)
Then I let u = 5 sin t which makes sqrt (25 - u^2) = 5cos t
du = 5 cos t dt
so then S du/ sqrt(25 - u^2) = S 5 cos t dt / 5 cos t
is this right???
= S dt = t + C
= sin ^-1 (u/5) +C = sin^-1 ((x+2)/5) + C
I find the 5 sin t = u where t = sin^-1 u/5 so if this is wrong let me know.