Math 2565, Spring 2020
Integrals. So many more integrals.
January 21, 2020
| Sean Fitzpatrick |
|---|
| University of Lethbridge |
 |
Trig substitution
Warm-Up
Choose a trig or hyperbolic substitution:
\begin{equation*}
\int \frac{x^2}{\sqrt{x^2+4}}\,dx
\end{equation*}
\begin{equation*}
\int\frac{\sqrt{4-x^2}}{x^2}\,dx
\end{equation*}
Integrals involving \(a^2-x^2\)
Examples:
-
\(\di\int \sqrt{9-x^2}\,dx\)
-
\(\di \int_{-3}^3\sqrt{9-x^2}\,dx\)
- \(\displaystyle \int_{-1}^1x^2\sqrt{1-x^2}\,dx\)
Integrals involving \(a^2+x^2\)
Examples:
-
\(\di \int\sqrt{4+x^2}\,dx\)
-
\(\di\int\frac{x^2}{\sqrt{x^2+3}}\,dx\)
-
\(\di \int\frac{x}{(x^2+9)^{3/2}}\,dx\)
Integrals involving \(x^2-a^2\)
Examples:
-
\(\di \int \sqrt{x^2-16}\,dx\)
-
\(\di \int_4^8 \sqrt{x^2-16}\,dx\)
- \(\displaystyle \int \frac{\sqrt{x^2-1}}{x}\,dx\)
Partial Fractions
Distinct linear factors
Look for decomposition
\begin{equation*}
\frac{\text{numerator goes here}}{(x-c_1)(x-c_2)\cdots (x-c_k)} = \frac{A_1}{x-c_1}+\frac{A_2}{x-c_2}+\cdots + \frac{A_k}{x-c_k}\text{.}
\end{equation*}
Examples:
\begin{equation*}
\int \frac{2x+1}{x^2-5x+6}\,dx \quad \quad \int \frac{x^2+4}{x^3-4x}\,dx
\end{equation*}
Repeated linear factors
For a factor \((x-a)^k\) in the denominator, need one term for each power:
\begin{equation*}
\frac{\text{numerator of some sort}}{(x-a)^k} = \frac{A_1}{x-a}+\frac{A_2}{(x-a)^2}+\cdots + \frac{A_k}{(x-a)^k}\text{.}
\end{equation*}
Examples:
\begin{equation*}
\int \frac{3x+20}{(x+8)^2}\,dx \quad \quad \int \frac{9x^2+11x+7}{x(x+1)^2}\,dx
\end{equation*}
Quadratic factors
For irreducible quadratics, need terms of form
\begin{equation*}
\frac{Ax+B}{x^2+ax+b}\text{.}
\end{equation*}
Example:
\begin{equation*}
\int\frac{2x^2+x+1}{(x+1)(x^2+9)}\,dx\text{.}
\end{equation*}
With long division
Partial fractions only works if degree of numerator is less than the denominator. Examples:
\begin{equation*}
\int \frac{x^3}{x^2-x-6}\,dx \quad \quad \int \frac{x^2+1}{x^2-1}\,dx\text{.}
\end{equation*}