How to integrate the following ??

intx/((x + 2)sqrt(x + 1)) dx

1 Answer
Oct 1, 2017

int x/((x+2)sqrt(x+1)) dx = 2sqrt(x+1) -4arctansqrt(x+1) +C

Explanation:

Substitute t=sqrt(x+1), dt = dx/(2sqrt(x+1)), x= t^2-1 so that:

int x/((x+2)sqrt(x+1)) dx = 2int (t^2-1)/(t^2+1)dt

Split now the numerator to simplify:

2int (t^2-1)/(t^2+1)dt = 2int (t^2+1-2)/(t^2+1)dt = 2int dt - 4 int dt/(t^2+1)

Both integrals can now be solved directly:

int x/((x+2)sqrt(x+1)) dx = 2t -4arctant +C

and undoing the substitution:

int x/((x+2)sqrt(x+1)) dx = 2sqrt(x+1) -4arctansqrt(x+1) +C