# Section 2.4 Calculus 2

Integrating with Partial Fractions

## 2.4 Integrating with Partial Fractions

### 2.4.1 Rational Functions and Partial Fractions

• A function of the form $$\frac{f(x)}{g(x)}$$ where $$f,g$$ are both polynomials is called rational.
• The rational function $$\frac{f(x)}{(x+r)^m}$$ may be split into the partial fractions $$\frac{A_1}{x+r}+\frac{A_2}{(x+r)^2}+\dots+\frac{A_m}{(x+r)^m}$$, provided the degree of the numerator is less than the denominator.
• Example Expand $$\frac{2x^2-7x+6}{(x-2)^3}$$ using partial fractions.
• The rational function $$\frac{f(x)}{(x^2+px+q)^n}$$ (where $$x^2+px+q$$ is irreducible) may be split into the partial fractions $$\frac{B_1x+C_1}{x^2+px+q}+ \frac{B_2x+C_2}{(x^2+px+q)^2}+ \dots+ \frac{B_mx+C_m}{(x^2+px+q)^n}$$, provided the degree of the numerator is less than the denominator.
• Example Expand $$\frac{3x^2+2x+4}{x^4+2x^2+1}$$ using partial fractions.
• When the denominator is a product of $$(x+r)^m$$ and $$(x^2+px+q)^n$$ terms, simply sum up the appropriate partial fractions for each factor.
• Example Describe the partial fractions which expand the rational function $$\frac{f(x)}{(x+3)^3(x^2-2x+3)^2}$$.

### 2.4.2 Integrating Partial Fractions

• Expanding rational functions using partial fractions allows us to integrate.
• Example Find $$\int\frac{2x^2+5x-9}{(x-1)(x+1)(x-2)}\,dx$$.
• Example Find $$\int\frac{4y^2+14y+15}{y^3+4y^2+5y}\,dy$$.
• If the numerator has degree greater than or equal to the denominator, you will need to use long polynomial division to break down the rational function first.
• Example Find $$\int\frac{2t^3+t^2+3t+2}{(1+t)(1+t^2)}\,dt$$.

### Review Exercises

1. Expand $$\frac{4x^2+16x+17}{(x+2)^3}$$ using partial fractions.
2. Expand $$\frac{-y^2+2y-4}{(y^2+4)^2}$$ using partial fractions.
3. Expand $$\frac{3r^3+r^2+3}{r^4+3r^2}$$ using partial fractions.
4. Find $$\int\frac{3z+2}{z^2+2z+1}\,dz$$.
5. Find $$\int\frac{3x^2+35}{x^3+5x}\,dx$$.
6. Find $$\int\frac{2v^3+4v^2+4v+2}{v^2+2v}\,dv$$.
7. Find $$\int\frac{2x^3+6x^2+4x+2}{(x+1)^2(x^2+1)}\,dx$$.
8. Describe the expansion of $$\frac{f(t)}{(t+1)^2(t^2+9)}$$ using partial fractions. (Assume $$f(t)$$ is a polynomial of degree less than 4.)
9. Find $$\int \frac{-x^2+6x-3}{(x+3)(x^2+1)}\,dx$$.

Solutions 1-4

Solutions 5-9

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 8.4