How to integrate the following ??

Question

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

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Answer 1 · 2017-10-01T16:02:09

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

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#