Question

The value of `int_1^3 dx/(x(1 + x^2))` is ______ 

Options

• `1/2log(9/5)`

• `1/2log(5/9)`

• `1/2log(4/9)`

• `1/2log(9/4)`

Answer 1

The value of `int_1^3 dx/(x(1 + x^2))` is `underline(1/2log(9/5))`.

Explanation:

`int_1^3 dx/(x(1 + x^2)) = int_1^3 (1/x - 1/(1 + x^2))dx`

= `int_1^3 1/x dx - 1/2int_1^3 (2x)/(1 + x^2)dx` 

= `[logx]_1^3 - 1/2[log(1 + x^2)]_1^3`

= `log3 - log1 - 1/2(log10 - log2)`

= `log3 - 1/2log5`

= `1/2log3^2 - 1/2 log5`

= `1/2(log9 - log5)`

= `1/2log(9/5)`