Question

`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.

Options

• `1/4log(2x^2 - 2x + 3) - sqrt(5)/2 tan^-1((2x - 1)/sqrt(5)) + C`

• `3/4log(2x^2 - 2x + 3) - sqrt(5)/2 tan^-1((2x - 1)/sqrt(5)) + C`

• `3/4log(2x^2 - 2x + 3) - sqrt(5)/2 tan^-1((4x - 2)/sqrt(5)) + C`

• `1/4log(2x^2 - 2x + 3) - sqrt(5)/2 tan^-1((4x - 2)/sqrt(5)) + C`

Answer 1

`int(3x + 1)/(2x^2 - 2x + 3)dx` equals `underlinebb(3/4log(2x^2 - 2x + 3) - sqrt(5)/2 tan^-1((2x - 1)/sqrt(5)) + C)`.

Explanation:

I = `int(3x + 1)/(2x^2 - 2x + 3)dx`

= `3/4int(4x - 2)/(2x^2 - 2x + 3)dx + 5/2int1/(2x^2 - 2x + 3)dx`

= `3/4log(2x^2 - 2x + 3) + 5/4int1/((x - 1/2)^2 + 5/4)dx`

= `3/4log(2x^2 - 2x + 3) + sqrt(5)/2tan^-1((2x - 1)/sqrt(5)) + c`