How do you integrate 2 / (x^3 + 1) using partial fractions?
1 Answer
Mar 14, 2018
Factorize the denominator then apply partial fraction decomposition.
Explanation:
Let
I=int2/(x^3+1)dx
Factorize the denominator:
I=2int1/((x+1)(x^2-x+1))dx
Apply partial fraction decomposition:
I=2/3int(1/(x+1)-(x-2)/(x^2-x+1))dx
Factor out the term where the numerator is a multiple of the derivative of the denominator:
I=2/3int(1/(x+1)-1/2* (2x-1)/(x^2-x+1)+3/2*1/(x^2-x+1))dx
Complete the square in the denominator:
I=2/3int(1/(x+1)-1/2*(2x-1)/(x^2-x+1)+6/((2x-1)^2+3))dx
Integrate term by term:
I=2/3{ln|x+1|-1/2ln|x^2-x+1|+sqrt3tan^(-1)((2x-1)/sqrt3)}+C
Simplify:
I=ln|x+1|-1/3ln|x^3+1|+2/sqrt3tan^(-1)((2x-1)/sqrt3)+C