Question

`int (2x)/((2 + x^2)(3 + x^2)) dx` = ______ 

Options

• log|(2 + x²) (3 + x²)| + c

• `log|(2 + x^2)/(3 + x^2)| + c`

• `log|(3 + x^2)/(2 + x^2)| + c`

• `log|(3 - x^2)/(2 - x^2)| + c`

Answer 1

`int (2x)/((2 + x^2)(3 + x^2)) dx` = `underline(log|(2 + x^2)/(3 + x^2)| + c)`

Explanation:

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

Put x² = t ⇒ 2x dx = dt

∴ I = `intdt/((2 + t)(3 + t))`

∴ I = `int(1/(2 + t) - 1/(3 + t))dx`

= `int dt/(2 + t) - intdt/(3 + t)`

= log|2 + t| - log|3 + t| + c

= log|2 + x²| - log|3 + x²| + c

∴ I = `log|(2 + x^2)/(3 + x^2)| + c`