How do you integrate int 1/(s+1)^21(s+1)2 using partial fractions?

2 Answers
Oct 26, 2017

The answer is =-1/(1+s)+C=11+s+C

Explanation:

You don't need partial fractions

Perform the substitution

u=1+su=1+s, du=dsdu=ds

int(ds)/(1+s)^2=int(u)^-2duds(1+s)2=(u)2du

=-1/(u)=1u

=-1/(1+s)+C=11+s+C

Oct 26, 2017

Partial fraction decomposition does not help; it gives you: 1/(s+1)^21(s+1)2

The integral int 1/(s+1)^2ds1(s+1)2ds is best integrated by "u" substitution.

Explanation:

Set up the expansion equation:

1/(s+1)^2 = A/(s+1)+B/(s+1)^21(s+1)2=As+1+B(s+1)2

Multiply both sides by (s+1)^2(s+1)2:

1 = A(s+1)+B1=A(s+1)+B

A = 0, B=1A=0,B=1

The decomposition is the same as the original:

1/(s+1)^2 = 1/(s+1)^21(s+1)2=1(s+1)2

Returning to the integral:

int 1/(s+1)^2 ds1(s+1)2ds

let u = s+1u=s+1, then du = dsdu=ds and the integral becomes:

int u^-2 du = -u^-1 + Cu2du=u1+C

Reverse the substitution:

int 1/(s+1)^2 ds = -(s+1)^-1 + C1(s+1)2ds=(s+1)1+C