Title: Math 2250 Integration Review

Q: [latex]
\begin{equation*}
\int x^n \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int \sin k x \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int \sin k x \, dx = -\frac{1}{k} \cos k x + C
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int \cos k x \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int \cos k x \, dx = \frac{1}{k} \sin k x + C.
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int \sec^2 k x \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int \sec^2 k x \, dx = \frac{1}{k} \tan k x + C.
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\csc^2 k x =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\csc^2 k x = - \frac{1}{k} \cot k x + C.
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int \sec kx \tan kx \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int \sec kx \tan kx \, dx = \frac{1}{k} \sec kx + C.
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int \csc kx \cot kx \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int \csc kx \cot kx \, dx = - \frac{1}{k} \csc kx + C.
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int e^{kx} \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int e^{kx} \, dx = \frac{1}{k} e^{kx} + C.
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int \frac{1}{x} \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int \frac{1}{x} \, dx = \ln |x| + C.
\end{equation*}
The absolute value might look weird here, but trying an example should convince you that it's right. If
\begin{equation*}
|x| = \begin{cases}
x & \text{if } x > 0 \\
-x & \text{if } x < 0 \\
\end{cases}
\end{equation*}
then for $x < 0$, we see that
\begin{equation*}
\frac{d}{dx} \ln |x| = \frac{d}{dx} \ln -x = -\frac{1}{-x} = \frac{1}{x}
\end{equation*}
by the chain rule (and $\ln |x|$ is defined for $x < 0$ while $\ln x$ isn't).
[/latex]

Q: [latex]
\begin{equation*}
\int \frac{1}{\sqrt{1 - k^2 x^2}} \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int \frac{1}{\sqrt{1 - k^2 x^2}} \, dx = \frac{1}{k} \arcsin k x + C.
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int \frac{1}{1 + k^2 x^2} \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int \frac{1}{1 + k^2 x^2} \, dx = \frac{1}{k} \arctan kx + C.
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int \frac{1}{x \sqrt{k^2 x^2 - 1}} \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int \frac{1}{x \sqrt{k^2 x^2 - 1}} \, dx = \operatorname{arcsec} kx + C.
\end{equation*}
as long as $kx > 1$.
[/latex]

Q: [latex]
\begin{equation*}
\int a^{kx} \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int a^{kx} \, dx = \frac{1}{k \ln a} a^{kx} + C.
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int_a^a f(x) \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int_a^a f(x) \, dx = 0
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int_a^b k f(x) \, dx =
\end{equation*}
[/latex]
A: [latex]
$$\int_a^b k f(x) \, dx = k \int_a^b f(x) \, dx.$$
[/latex]

Q: [latex]
\begin{equation*}
\int_a^b f(x) + g(x) \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int_a^b f(x) + g(x) \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx.
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int_a^b f(x) \, dx + \int_b^c f(x) \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int_a^b f(x) \, dx + \int_b^c f(x) \, dx = \int_a^c f(x) \, dx.
\end{equation*}
[/latex]

Q: [latex]
If $f(x)$ is integrable on $[a,b]$, the \emph{average value of $f(x)$ over $[a,b]$} or \emph{mean value of $f(x)$ over $[a,b]$} is
[/latex]
A: [latex]
\begin{equation*}
\operatorname{Mean}(f) = \frac{1}{b-a} \int_a^b f(x) \, dx.
\end{equation*}
[/latex]

Q: [latex]
(The $u$-substitution rule) If $u = g(x)$, then
\begin{equation*}
\int f(g(x)) g'(x) \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int f(g(x)) g'(x) \, dx = \int f(u) \, du.
\end{equation*}
[/latex]