Partial Fractions

Irreducible quadratic factors are quadratic factors that when set equal to zero only have complex roots. As a result they cannot be reduced into factors containing only real numbers, hence the name irreducible. Examples include

$x^2+1$ or indeed

$x^2+a$ for any real number

$a>0$ ,

$x^2+x+1$ (use the quadratic formula to see the roots), and

$2x^2-x+1$ .

When

$Q(x)$ has irreducible quadratic factors, it affects our decomposition.

Irreducible quadratic factors of $Q(x)$

For every irreducible quadratic factor $x^2 + bx + c$ of $Q(x)$ , we have $\displaystyle\frac{Ax+B}{x^2+bx+c}$ in the decomposition.
For every repeated irreducible quadratic factor $(x^2+bx+c)^n$ of $Q(x)$ , we have the $n$ terms $\displaystyle\frac{A_1 x + B_1}{x^2+bx+c} + \frac{A_2x+B_2}{(x^2+bx+c)^2}+\ldots + \frac{A_nx+B_n}{(x^2+bx+c)^n}$ in the decomposition.

Example: We have the decomposition

$\displaystyle{ 2x^3+5x-1 \over (x+1)^3(x^2+1)^2 } = \displaystyle{ { A \over x+1 } + { B \over (x+1)^2 }+ { C \over (x+1)^3 } + { Dx+E \over x^2+1 } + { Fx+G \over (x^2+1)^2 } }.$
Finding the coefficients

$A$ ,

$B$ , etc. for these terms in the decomposition can be challenging. We can't just plug in the roots of

$Q(x)$ to get the coefficients one at a time, since the quadratic factors don't have real roots. We either have to plug in lots of different values of

$x$ , or compare the corresponding coefficients of

$1$ ,

$x$ ,

$x^2$ , etc. Either way, we get a system of linear equations to solve.

Finally, solving the integrals at the end is harder than with linear factors, but we can do it:

To integrate $\displaystyle\int \frac{x}{(x^2+a^2)^n}\,dx,$ we can substitute $u=x^2+a^2$ .

To integrate $\displaystyle\int \frac{dx}{(x^2+a^2)^n},$ where $u$ -substitution will not work, we can substitute $x=a\tan(\theta)$ .

These types of problems are explained in the following video.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Irreducible Quadratic Factors

Irreducible quadratic factors of Q(x)Q(x)

Irreducible quadratic factors of $Q(x)$