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Question
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Solution
Let I = `int (x + 7)/(x^2 + 4x + 7)dx`
On applying partial integration method
`x + 7 = "A" d/dx(x^2 + 4x + 7) + "B"`
x + 7 = A(2x + 4) + B
Then, A = `1/2` and B = 5
Then, I = `int(1/2(2x + 4) + 5)/(x^2 + 4x + 7)dx`
= `1/2 int ((2x + 4))/(x^2 + 4x + 7)dx + 5 int 1/((x^2 + 4x + 7))dx`
= `1/2 log |x^2 + 4x + 7| + 5 int 1/((x + 2)^2 + (sqrt(3))^2) dx + c`
= `1/2 log |x^2 + 4x + 7| + 5/sqrt(3) tan^-1((x + 2)/sqrt(3)) + c`
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