Example:
I = ∫ dx /
(x2 + 2x + 5)
Complete the square with
(x + 1)2 +
4 = (x + 1)2 +
22
I
= ∫ dx / [ (x + 1)2 +
22 ]
Now let u
= x + 1, du
= dx
I
= ∫ du / ( u2 + 22
) which suggests the following trig
substitution
Let u = 2 tan θ, du
= 2 sec2
θ d θ, u2 + 22 =
22 sec2 θ
I
= ∫ 2 sec2
θ d θ / 22 sec2
θ = ∫ (1/2) d θ =
(1/2) θ + C
Now
θ = tan-1 (u/2) =
tan-1 [( x+1)/2]
So the integral becomes:
I =
(1/2) tan-1 [( x+1)/2]
+ C (result)
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