Question

Need some help with indefinite integrals pleasee

Answer 1

Observe that
`(x^2+2x+2)'=2x+2`.

You have that
`\int (2x)/(x^2+2x+2)\; dx=\int (2x+2-2)/(x^2+2x+2)\; dx=\int (2x+2)/(x^2+2x+2)\; dx+\int (1)/(x^2+2x+2)\; dx=I_1+2I_2`.

To compute
`I_1` you set
`y=x^2+2x+2`, so
`dy=2x+2\; dx`. Therefore,
`I_1=\int (1)/(y)\; dy=\ln |y|=\ln |x^2+2x+2|=\ln (x^2+2x+2)`.

For
`I_2` observe that
`x^2+2x+2=(x+1)^2+1`. Let
`y=x+1`. Then
`dy=dx` and
`I_2=\int (1)/(y^2+1)=\arctan y=\arctan (x+1)`.

So,
`\int (2x)/(x^2+2x+2)=\ln (x^2+2x+2)+2\arctan (x+1)+c`.