Evaluate the following integral: `int_0^(1//sqrt(3))tan^(-1)((3x-x^3)/(1-3x^2))dx`

Explanation

Given:`int_0^(1/sqrt(3))tan^(-1)((3x-x^3)/(1-3x^2))dx`

`I=int_0^(1/sqrt(3))tan^(-1)((3x-x^3)/(1-3x^2))dx`

Let `x=tant`

`I=int_0^(1/sqrt(3))tan^(-1)((3tant-tan^3t)/(1-3tan^2t))dx`

`I=int_0^(1/sqrt(3))tan^(-1)(tan3t)dx`

`I=int_0^(1/sqrt(3)) 3tdx`

`I=int_0^(1/sqrt(3)) 3tan^(-1)xdx`

`I=3int_0^(1/sqrt(3)) tan^(-1)xdx`

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