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What is the integral `int (2+3*x)/(x^2+1) dx`

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What is the integral `int (2+3*x)/(x^2+1) dx`

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The integral `int (2+3*x)/(x^2+1) dx` has to be determined.

`int (2+3*x)/(x^2+1) dx`

=> `int 2/(x^2+1) dx + int 3*x/(x^2 + 1) dx`

=> `2*int 1/(x^2+1) dx + 3*int x/(x^2 + 1) dx`

=> `2*tan^-1x + 3*int x/(x^2 + 1) dx`

let `y = x^2 + 1` , `x*dx = (1/2)dy`

=> `2*tan^-1x + (3/2)*int (1/y) dy`

=> `2*tan^-1x + (3/2)*ln y`

substitute `y = x^2 + 1`

=> `2*tan^-1x + (3/2)*ln (x^2 + 1) + C`

The integral`int (2+3*x)/(x^2+1) dx`= `2*tan^-1x + (3/2)*ln (x^2 + 1) + C`

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