Question

∫ (4+x)² / (x²+4) please help me solve this? :(

∫ ((4+x)² / (x²+4))

Answer 1

The answer is `=x+4ln(x^2+4)+6arctan(x/2)+C`

Explanation:

We need

`int(dx)/(x^2+1)=arctan(x)+C`

Perform a polynomial long division

`(4+x)^2/(x^2+4)=(x^2+8x+16)/(x^2+4)=1+(8x+12)/(x^2+4)`

`=1+(8x)/(x^2+4)+12/(x^2+4)`

Therefore,

`int((4+x)^2dx)/(x^2+4)=int1dx+4int(2xdx)/(x^2+4)+12int(dx)/(x^2+4)`dx

`=x+4ln(x^2+4)+12int(dx)/(4((x/2)^2+1))`

`=x+4ln(x^2+4)+3int(dx)/((x/2)^2+1)`

`=x+4ln(x^2+4)+6arctan(x/2)+C`