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Question
`int (dx)/(x(x^2 + 1))` equals:
Options
`log |x| - 1/2 log |x^2 + 1| + C`
`log |x| + 1/2 log |x^2 + 1| + C`
`- log |x| + 1/2 log |x^2 + 1| + C`
`1/2 log |x| + log (x^2 + 1) + C`
Solution
`log |x| - 1/2 log |x^2 + 1| + C`
Explanation:
Let `I = int dx/(x (x^2 + 1))`
`= int x/(x (x^2 + 1)) dx`
Put x2 = t
2x dx = dt
`I = 1/2 int (2x dx)/(x (x^2 + 1))`
`= 1/2 int dt/(t (t + 1))`
Now, `1/(t (t + 1)) = A/t + B/(t + 1)`
1 = A(t + 1) + Bt
Putting t = 0, 1 = A
∴ A = 1
Putting t = -1, 1 = B(-1)
∴ B = -1
`therefore 1/(t (t + 1)) = 1/t - 1/(t + 1)`
`therefore 1/2 int 1/(t (t + 1)) dt = 1/2 int 1/t dt - 1/2 int 1/(t + 1) dt`
`= 1/2 log abs t - 1/2 log abs (t + 1) + C`
`= 1/2 log abs (x ^2) - 1/2 log abs(x ^2 + 1) + C`
`= log abs x - 1/2 log abs(x^2 + 1) + C`
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