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Question
Find: `int (x + 1)/((x^2 + 1)x) dx`
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Solution
Let `(x + 1)/((x^2 + 1)x) = (Ax + B)/(x^2 + 1) + C/x = ((Ax + B)x + C(x^2 + 1))/((x^2 + 1)x)`
⇒ `x + 1 = (Ax + B)x + C(x^2 + 1)` .....(An identity)
Equating the coefficients, we get
B = 1, C = 1, A + C = 0
Hence, A = –1, B = 1, C = 1
The given integral = `int (-x + 1)/(x^2 + 1) dx + int 1/x dx`
= `(-1)/2 int (2x - 2)/(x^2 + 1) dx + int 1/x dx`
= `(-1)/2 int (2x)/(x^2 + 1) dx + int 1/(x^2 + 1) dx + int 1/x dx`
= `(-1)/2 log(x^2 + 1) + tan^-1x + log |x| + c`
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