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Chapter 1
Integration Techniques

Integration by Parts

Choose $u$ using LIATE: Logarithm, Inverse trig, Algebraic, Trig, Exponential.

Trig Substitution

$\sqrt{a^2-x^2}$: let $x=a\sin\theta$.
$\sqrt{a^2+x^2}$: let $x=a\tan\theta$.
$\sqrt{x^2-a^2}$: let $x=a\sec\theta$.

Partial Fractions

Decompose rational functions. For repeated factor $(x-a)^n$, include terms $\frac{A_1}{x-a}+\cdots+\frac{A_n}{(x-a)^n}$.

Key Memorize

$\int \ln x\,dx = x\ln x - x + C$
$\int \sec^3 x\,dx = \tfrac{1}{2}\sec x\tan x + \tfrac{1}{2}\ln|\sec x+\tan x|+C$

$$\int u\,dv = uv - \int v\,du$$
▸ Example

Evaluate $\displaystyle\int x e^x\,dx$.

Let $u=x$, $dv=e^x dx$. Then $du=dx$, $v=e^x$.
Answer: $x e^x - e^x + C$
Chapter 2
Improper Integrals

Type I (Infinite Limits)

$\int_a^\infty f(x)\,dx = \lim_{t\to\infty}\int_a^t f(x)\,dx$. Converges if limit is finite.

$p$-Test for $\int_1^\infty x^{-p}dx$

Converges if $p>1$; Diverges if $p\le 1$.

$$\int_1^\infty \frac{1}{x^p}\,dx = \begin{cases}\dfrac{1}{p-1} & p>1\\[4pt] \text{diverges} & p\le 1\end{cases}$$
▸ Example

Does $\displaystyle\int_1^\infty \frac{1}{x^2}\,dx$ converge?

$p=2>1$, so it converges. Value: $\left[-\frac{1}{x}\right]_1^\infty = 0-(-1)=1$.
Chapter 3
Applications of Integration

Arc Length

$L=\displaystyle\int_a^b\sqrt{1+[f'(x)]^2}\,dx$

Volume – Disk/Washer

Disk: $V=\pi\displaystyle\int_a^b [f(x)]^2 dx$
Washer: $\pi\displaystyle\int_a^b\!\left([R(x)]^2-[r(x)]^2\right)dx$

Shell Method

$V=2\pi\displaystyle\int_a^b x\,f(x)\,dx$ (rotation about $y$-axis)

Surface Area

$S=2\pi\displaystyle\int_a^b f(x)\sqrt{1+[f'(x)]^2}\,dx$

Chapter 4
Sequences & Series

Geometric Series

$\displaystyle\sum_{n=0}^\infty ar^n = \frac{a}{1-r}$ if $|r|<1$; diverges if $|r|\ge 1$.

$p$-Series

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^p}$ converges iff $p>1$.

Ratio Test

$L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$: converges if $L<1$; diverges if $L>1$; inconclusive if $L=1$.

Alternating Series Test

$\sum(-1)^n b_n$ converges if (1) $b_n$ decreasing, (2) $b_n\to 0$.

$$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}, \quad \sum_{n=0}^\infty x^n = \frac{1}{1-x} \text{ for } |x|<1$$
Chapter 5
Power Series & Taylor Series

Radius of Convergence

For $\sum a_n(x-c)^n$: $R=\dfrac{1}{\limsup|a_n|^{1/n}}$. Use Ratio Test to find $R$.

Key Maclaurin Series

$e^x=\displaystyle\sum_{n=0}^\infty\frac{x^n}{n!}$,  $\sin x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^n x^{2n+1}}{(2n+1)!}$,
$\cos x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^n x^{2n}}{(2n)!}$

$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$$
▸ Example

Find the Maclaurin series for $e^{2x}$.

Replace $x$ with $2x$: $\displaystyle e^{2x}=\sum_{n=0}^\infty \frac{(2x)^n}{n!}=\sum_{n=0}^\infty \frac{2^n x^n}{n!}$
Chapter 6
Polar & Parametric Equations

Parametric Arc Length

$L=\displaystyle\int_\alpha^\beta\sqrt{\!\left(\frac{dx}{dt}\right)^2\!+\!\left(\frac{dy}{dt}\right)^2}\,dt$

Polar Area

$A=\dfrac{1}{2}\displaystyle\int_\alpha^\beta [r(\theta)]^2\,d\theta$

Conversion

$x=r\cos\theta$, $y=r\sin\theta$, $r^2=x^2+y^2$

Slope in Parametric

$\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$

Chapter 7
Differential Equations

Separable Equations

Rewrite as $g(y)\,dy = f(x)\,dx$, then integrate both sides.

Linear First-Order

$y'+P(x)y=Q(x)$. Integrating factor: $\mu=e^{\int P\,dx}$. Solution: $y=\dfrac{1}{\mu}\displaystyle\int\mu Q\,dx$.

$$\text{Separable: } \int \frac{dy}{g(y)} = \int f(x)\,dx$$
▸ Example

Solve $\dfrac{dy}{dx}=xy$.

$\dfrac{dy}{y}=x\,dx \Rightarrow \ln|y|=\dfrac{x^2}{2}+C \Rightarrow y=Ae^{x^2/2}$
Examination Questions

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