Evaluate: `int1/((x^2+2x+10)^2)dx`

Explanation

Let `I=int frac{1}{(x^{2}+2 x+10)^{2}} d x =int frac{1}{[(x+1)^{2}+3^{2}]^{2}} d x Let x+1=3 tan theta`

On differentiating both sides, we get

`d x=3 sec ^{2} theta d theta`

`therefore I=int frac{1}{[3^{2} tan ^{2} theta+3^{2}]^{2}} 3 sec ^{2} theta {d} theta`

`=frac{1}{27} int frac{sec ^{2} theta}{sec ^{4} theta} d theta`

`=frac{1}{27} int frac{1}{sec ^{2} theta} d theta`

`=frac{1}{27} int cos ^{2} theta d theta`

`=frac{1}{54} int(1+cos 2 theta) d theta`

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