How do you integrate 1x2+3x+2 using partial fractions?

1 Answer
Nov 30, 2016

1x2+3x+2dx=ln(x+1x+2)

Explanation:

Factorize the denominator:

x2+3x+2=(x+1)(x+2)

Now develop in partial fractions using parametric numerators:

1x2+3x+2=Ax+1+Bx+2

Expand and equate the coefficient of the same order of the left side and right side numerators to determine A and B:

1x2+3x+2=A(x+2)+B(x+1)(x+1)(x+2)=Ax+2A+Bx+B(x+1)(x+2)=(A+B)x+(2A+B)(x+1)(x+2)

So:

A+B=0
2A+B=1

Solving the system:

A=1
B=1

Finally:

1x2+3x+2=1x+11x+2

We are now ready to integrate:

1x2+3x+2dx=(1x+11x+2)dx=

=dxx+1dxx+2=ln(x+1)ln(x+2)=ln(x+1x+2)