Title: Math 2260 Chapter 8 (Methods of Integration)

Q: The integration by parts formula is
[$$]\int u \, dv = [/$$]
A: [latex]
\begin{center}
\begin{equation*}
\int u \, dv = uv - \int v \, du.
\end{equation*}
\end{center}
[/latex]

Q: The integration by parts formula is
[$$] \int f(x) g'(x) \, dx = [/$$]
A: [latex]
\begin{huge}
\begin{multline*} \int f(x) g'(x) \, dx = \\
f(x) g(x) - \int f'(x) g(x) \, dx.
\end{multline*}
\end{huge}
[/latex]

Q: The integration by parts formula for definite integrals is
[$$] \int_a^b f(x) g'(x) \, dx = [/$$]
A: [latex]
\begin{huge}
\begin{multline*} \int_a^b f(x) g'(x) \, dx = \\
f(x) g(x) \bigg|_{x=a}^b - \int_a^b f'(x) g(x) \, dx.
\end{multline*}
\end{huge}
[/latex]

Q: How to integrate
[$$] \int \sin^{2k+1} x \cos^n x \, dx [/$$]
A: [latex]
\begin{huge}
Use
\begin{equation*}
\sin^{2k} x = (1 - \cos^2 x)^{k}
\end{equation*}
to rewrite the integral as
\begin{equation*}
\int (1 - \cos^2 x)^k \cos^n x \sin x\, dx
\end{equation*}
and do the $u$-substitution $u = \cos x$.
\end{huge}
[/latex]

Q: [latex]
\begin{equation*}
\int \sin mx \sin nx =
\end{equation*}
[/latex]
A: [latex]
\begin{huge}
\begin{multline*}
\int \sin mx \sin nx = \\
\frac{1}{2} \left[ \cos(m-n) x - \cos(m+n) x \right]
\end{multline*}
\end{huge}
[/latex]

Q: [latex]
\begin{equation*}
\int \sin mx \cos nx =
\end{equation*}
[/latex]
A: [latex]
\begin{huge}
\begin{multline*}
\int \sin mx \cos nx = \\
\frac{1}{2} \left[ \sin(m-n) x + \sin(m+n) x \right]
\end{multline*}
\end{huge}
[/latex]

Q: [latex]
\begin{equation*}
\int \cos mx \cos nx =
\end{equation*}
[/latex]
A: [latex]
\begin{huge}
\begin{multline*}
\int \cos mx \cos nx = \\
\frac{1}{2} \left[ \cos(m-n) x + \cos(m+n) x \right]
\end{multline*}
\end{huge}
[/latex]

Q: To integrate
[$$] \int \tan^m \! x \, \sec^{2n} \! x \, dx[/$$]
(when [$]n[/$] is not zero)
A: [latex]
\begin{huge}
Use $ \sec^2 x = \tan^2 x + 1 $ to write
\begin{multline*} \int \tan^m \!x \, \sec^{2n} \! x \, dx = \\
\int \tan^m \! x \, (\tan^2 \! x + 1)^{n-1} \sec^2 x \, dx.
\end{multline*}
and substitute $u = \tan x$, $du = \sec^2 \! x \, dx$.
\end{huge}
[/latex]

The trig substitution for integrals involving [$]\sqrt{a^2 + x^2}[/$]

The trig substitution for integrals involving [$]\sqrt{a^2 - x^2}[/$]

The trig substitution for integrals involving [$]\sqrt{x^2 - a^2}[/$]

Q: [latex]
\begin{huge}
If $f(x)$ is a polynomial of degree $< m$, the partial fraction decomposition of
\begin{center}
\begin{equation*}
\frac{f(x)}{(x - r)^m} = \dots
\end{equation*}
\end{center}
\end{huge}
[/latex]
A: [latex]
\begin{huge}
\begin{equation*} \frac{f(x)}{(x - r)^m} = \frac{c_1}{x-r} + \frac{c_2}{(x-r)^2} + \cdots \frac{c_m}{(x-r)^m}.
\end{equation*}
where $c_1, \dots, c_m$ are real numbers.
\end{huge}
[/latex]

Q: How to integrate
[$$] \int \sin^{m} \! x \, \cos^{2k+1} \! x \, dx [/$$]
A: [latex]
\begin{huge}
Use
\begin{equation*}
\cos^{2k} x = (1 - \sin^2 x)^{k}
\end{equation*}
to rewrite the integral as
\begin{equation*}
\int \sin^m \!x \, (1 - \sin^2 x)^k \cos x\, dx
\end{equation*}
and do the $u$-substitution $u = \sin x$.
\end{huge}
[/latex]

Q: [latex]
\begin{large}
\begin{center}
To integrate $ \int \tan^m \! x \, dx$
\end{center}
\end{large}
[/latex]
A: [latex]
\begin{huge}
Use $\tan^2 x = \sec^2 x - 1$ to write
\begin{equation*} \int \tan^m \!x \, dx =
\int \tan^{m-2} \! x \, \sec^2 x \, dx - \int \tan^{m-2} \! x \, dx.
\end{equation*}
Substitute $u = \tan x$, $du = \sec^2 \! x \, dx$ in first integral and repeat procedure on second.
\end{huge}
[/latex]

Q: [latex]
\begin{center}
\begin{large}
To integrate
$\int \sec^m \! x \, dx$
\end{large}
\end{center}
[/latex]
A: [latex]
\begin{huge}
Integrate by parts with
\begin{equation*}
u = \sec^{m-2}\!x \quad dv = \sec^2 x \quad (v = \tan x)
\end{equation*}
to get a better integral in powers of $\sec x$ and $\tan x$.
\end{huge}
[/latex]

Q: [latex]
\begin{huge}
\begin{center}
If $f(x)$ is a polynomial of degree $< m$, the partial fraction decomposition of
\begin{equation*}
\frac{f(x)}{(x^2 +p x + q)^m}
\end{equation*}
\end{center}
where $x^2 + px + q$ has no real roots, is...
\end{huge}
[/latex]
A: [latex]
\begin{large}
\begin{multline*} \frac{f(x)}{(x^2 +p x + q)^m} = \\
\frac{c_1 x + c_2}{x^2 + px + q} + \frac{c_3 x + c_4}{(x^2 + px + q)^2} + \cdots + \frac{c_{2m-1} x + c_{2m}}{(x^2 + px + q)^m}.
\end{multline*}
where $c_1, \dots, c_{2m}$ are real numbers.
\end{large}
[/latex]

Q: [latex]
\begin{huge}
If $f(x)$ is a polynomial of degree $< m$, and the roots $r_i$ are \textbf{real} and \textbf{distinct} the partial fraction decomposition of
\begin{center}
\begin{equation*}
\frac{f(x)}{(x - r_1) \cdots (x - r_m)} = \dots
\end{equation*}
\end{center}
\end{huge}
[/latex]
A: [latex]
\begin{huge}
\begin{multline*}
\frac{f(x)}{(x - r_1) \cdots (x-r_m)} = \\
\frac{c_1}{x-r_1} + \frac{c_2}{x-r_2} + \cdots + \frac{c_m}{x-r_m}.
\end{multline*}
where $c_1, \dots, c_m$ are numbers.
\end{huge}
[/latex]

Q: [latex]
\begin{large}
If
\begin{align*}
x_0 &= a, \, x_1 = a + h, \dots, x_n = a + nh = b, \\
y_0 &= f(x_0), y_1 = f(x_1), \dots, y_n = f(x_n).
\end{align*}
the \emph{trapezoid rule} says
\begin{equation*}
\int_a^b f(x) \, dx \simeq
\end{equation*}
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*}
\int_a^b f(x) \, dx \simeq
  \frac{h}{2} (y_0 + 2 y_1 + 2 y_2 + \cdots + 2 y_{n-1} + y_n).
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
If $n$ is \emph{even} and
\begin{align*}
x_0 &= a, \, x_1 = a + h, \dots, x_n = a + nh = b, \\
y_0 &= f(x_0), y_1 = f(x_1), \dots, y_n = f(x_n).
\end{align*}
then \emph{Simpson's rule} says
\begin{equation*}
\int_a^b f(x) \, dx \simeq
\end{equation*}
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*}
\int_a^b f(x) \, dx
  \simeq \frac{h}{3} (y_0 + 4 y_1 + 2 y_2 + \cdots + 2 y_{n-2} + 4 y_{n-1} + y_n).
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{huge}
The error estimate for the \emph{trapezoid rule}
\end{huge}
[/latex]
A: [latex]
\begin{huge}
If $M \geq |f''(x)|$ on $[a,b]$, the error $E_t$ in computing the integral of $f(x)$ over $[a,b]$ using the trapezoid rule with $n$ intervals is bounded by
\begin{equation*}
|E_t| \leq \frac{M(b-a)^3}{12 n^2}
\end{equation*}
\end{huge}
[/latex]

Q: [latex]
\begin{huge}
The error estimate for \emph{Simpson's rule}
\end{huge}
[/latex]
A: [latex]
\begin{huge}
If $M \geq |f^{(4)}(x)|$ on $[a,b]$, the error $E_s$ in computing the integral of $f(x)$ over $[a,b]$ using Simpson's rule with $n$ intervals is bounded by
\begin{equation*}
|E_s| \leq \frac{M(b-a)^5}{180 n^4}
\end{equation*}
\end{huge}
[/latex]

Q: [latex]
\begin{large}
The improper integral
$\int_a^\infty f(x) \, dx$
\end{large}[/latex]
A: [latex]
\begin{huge}
\begin{equation*}
\int_a^\infty f(x) \, dx = \lim_{b \rightarrow \infty} \int_a^b f(x) \, dx.
\end{equation*}
If the limit doesn't exist, we say the improper integral \emph{diverges}.
\end{huge}
[/latex]

Q: [latex]
\begin{large}
The improper integral
$\int_{-\infty}^b f(x) \, dx$
\end{large}[/latex]
A: [latex]
\begin{huge}
\begin{equation*}
\int_{-\infty}^b f(x) \, dx = \lim_{a \rightarrow -\infty} \int_a^b f(x) \, dx.
\end{equation*}
If the limit doesn't exist, we say the improper integral \emph{diverges}.
\end{huge}
[/latex]

Q: [latex]
\begin{large}
The improper integral
$\int_{-\infty}^\infty f(x) \, dx$
\end{large}[/latex]
A: [latex]
\begin{huge}
\begin{equation*}
\int_{-\infty}^\infty f(x) \, dx = \int_{-\infty}^c f(x)\, dx + \int_c^\infty f(x) \, dx
\end{equation*}
where $c$ is any real number.
\end{huge}
[/latex]

Q: [latex]
\begin{huge}
\emph{Direct} comparison test (improper integrals).
If $0 \leq f(x) \leq g(x)$ on $[a,\infty)$ and
\begin{equation*}
\int_a^\infty g(x) \, dx
\end{equation*}
converges then
\end{huge}[/latex]
A: [latex]
\begin{large}
\begin{equation*}
\int_a^\infty f(x) \, dx \text{ converges }
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{huge}
\emph{Direct} comparison test (improper integrals).
If $0 \leq f(x) \leq g(x)$ on $[a,\infty)$ and
\begin{equation*}
\int_a^\infty f(x) \, dx
\end{equation*}
diverges then
\end{huge}[/latex]
A: [latex]
\begin{large}
\begin{equation*}
\int_a^\infty g(x) \, dx \text{ diverges }
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{huge}
\emph{Limit} comparison test (improper integrals).
If $0 < f(x)$ and $0<g(x)$ on $[a,\infty)$ and
\begin{equation*}
\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = L, 0 < L < \infty
\end{equation*}
then
\end{huge}[/latex]
A: [latex]
\begin{large}
The improper integrals
\begin{equation*}
\int_a^\infty f(x) \, dx \text{ and }
\int_a^\infty g(x) \, dx
\end{equation*}
\emph{both converge} or \emph{both diverge}.
\end{large}
[/latex]