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`

• `log |x| + 1/2 log |x^2 + 1| + c`

Answer 1

`log |x| - 1/2 log |x^2 + 1| + c`

Explanation:

I = `int (dx)/(x(x^2 + 1)) = int x/(x^2(x^2 + 1)) dx`

Put `x^2 = t, 2x dx = dt`

I = `1/2 int (2x dx)/(x^2(x + 1)) = 11/2 int (dt)/(t(t + 1)`

Now, `1/(t(t + 1)) = A/t + B/(t + 1)`

1 = A(t + 1) + Bt

Put t = 0, 1 = A

t = – 1, 1 = B(– 1)    ∴ B = – 1

∴ `1/(t(t + 1)) = 1/t - 1/(t + 1)` 

∴ `1/2 int 1/(t(t + 1)) dt = 1/2 int  1/t dt - 1/2 int 1/(t + 1) dt`

= `1/2 log |t| - 1/2 log |t + 1| + c`

= `1/2 log |x^2| - 1/2 log |t + 1| + c`

= `log |x| - 1/2 log|x^2 + 1| + c`