http://www.ck12.org Chapter 7. Integration Techniques
- For each distinct factorax+b,the right side must include a term of the form
A
ax+b.- For each repeated factor(ax+b)n,the right side must include n terms of the form
A 1
(ax+b)+A 2
(ax+b)^2 +A 3
(ax+b)^3 +...+An
(ax+b)n.Example 2:
Use the method of partial fractions to evaluate∫(xx++ 21 ) 2 dx.
Solution:
According to the guide above (item #3), we must assign the sum ofn=2 partial sums:
x+ 1
(x+ 2 )^2 =A
(x+ 2 )+B
(x+ 2 )^2.Multiply both sides by(x+ 2 )^2 :
x+ 1 =A(x+ 2 )+B
x+ 1 =Ax+( 2 A+B).Equating the coefficients of like terms from both sides,
1 =A
1 = 2 A+B.
Thus
A= 1.
B=− 1.
Therefore the partial fraction decomposition is
x+ 1
(x+ 2 )^2 =1
x+ 2 −1
(x+ 2 )^2.The integral will become