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Calculus 2
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Core Concepts & Formulas

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01 Integration Techniques ▼
Integration by Parts
\[\int u\,dv = uv - \int v\,du\]
★ LIATE rule for choosing \(u\): Logarithm · Inverse trig · Algebraic · Trig · Exponential
Trigonometric 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 \(\dfrac{P(x)}{Q(x)}\) into sums of simpler fractions, matching degree of numerator < denominator.
Worked Example
Evaluate \(\displaystyle\int x\,e^x\,dx\).
Let \(u=x\), \(dv=e^x dx\) → \(du=dx\), \(v=e^x\).
\(\displaystyle\int x e^x dx = xe^x - \int e^x dx = xe^x - e^x + C = e^x(x-1)+C\)
✓ Answer: \(e^x(x-1)+C\)
IBPTrig SubPartial Fractions
02 Improper Integrals ▼
Definition
\[\int_a^{\infty} f(x)\,dx = \lim_{t\to\infty}\int_a^t f(x)\,dx\]
★ \(\displaystyle\int_1^{\infty}\frac{1}{x^p}dx\) converges iff \(p>1\). Diverges if \(p\le 1\).
Key Test: p-integral
\[\int_1^{\infty}\frac{dx}{x^p} = \begin{cases}\frac{1}{p-1} & p>1 \\ \text{diverges} & p\le 1\end{cases}\]
Does \(\displaystyle\int_1^{\infty}\frac{1}{x^2}dx\) converge?
\(\lim_{t\to\infty}\left[-\frac{1}{x}\right]_1^t = \lim_{t\to\infty}\left(-\frac{1}{t}+1\right)=1\)
✓ Yes, converges to 1.
03 Sequences & Series ▼
Convergence Tests
Ratio Test: \(L=\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|\). Converges if \(L<1\), diverges if \(L>1\), inconclusive if \(L=1\).
Integral Test: If \(f\) is positive, decreasing, continuous on \([1,\infty)\): \(\sum a_n\) converges iff \(\int_1^\infty f(x)\,dx\) converges.
★ Geometric series: \(\displaystyle\sum_{n=0}^{\infty}ar^n = \frac{a}{1-r}\) for \(|r|<1\).
★ Alternating Series Test: \(\sum(-1)^n b_n\) converges if \(b_n\searrow 0\).
Find the sum: \(\displaystyle\sum_{n=0}^{\infty}\left(\frac{2}{3}\right)^n\).
\(a=1,\ r=\frac{2}{3},\ |r|<1\). Sum \(=\dfrac{1}{1-2/3}=3\).
✓ Sum = 3
04 Power Series & Taylor/Maclaurin Series ▼
Taylor Series
\[f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n\]
★ Must-know Maclaurin series:
\(e^x=\sum\frac{x^n}{n!}\)  ·  \(\sin x=\sum\frac{(-1)^n x^{2n+1}}{(2n+1)!}\)  ·  \(\cos x=\sum\frac{(-1)^n x^{2n}}{(2n)!}\)
\(\dfrac{1}{1-x}=\sum x^n,\ |x|<1\)  ·  \(\ln(1+x)=\sum\frac{(-1)^{n+1}x^n}{n},\ |x|\le 1\)
Radius of Convergence
Use Ratio Test on \(\sum c_n(x-a)^n\): \(R=\dfrac{1}{\lim|c_{n+1}/c_n|}\)
Find the Maclaurin series for \(e^{-x^2}\) up to \(x^4\).
Replace \(x\) with \(-x^2\) in \(e^x=1+x+\frac{x^2}{2}+\cdots\):
\(e^{-x^2}=1-x^2+\dfrac{x^4}{2}-\cdots\)
✓ \(1-x^2+\frac{x^4}{2}-\cdots\)
05 Parametric & Polar Curves ▼
Parametric Derivatives & Arc Length
\(\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}\)  ·  \(L=\displaystyle\int_\alpha^\beta\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt\)
Polar Area
\[A=\frac{1}{2}\int_\alpha^\beta r^2\,d\theta\]
★ Convert: \(x=r\cos\theta,\ y=r\sin\theta,\ r^2=x^2+y^2,\ \tan\theta=y/x\)
Find the slope of \(x=t^2,\ y=t^3\) at \(t=2\).
\(\dfrac{dy}{dx}=\dfrac{3t^2}{2t}=\dfrac{3t}{2}\). At \(t=2\): slope \(=3\).
✓ Slope = 3
06 Applications of Integration ▼
Volumes of Revolution
Disk/Washer: \(V=\pi\displaystyle\int_a^b[R(x)^2-r(x)^2]\,dx\)

Shell: \(V=2\pi\displaystyle\int_a^b x\,f(x)\,dx\)
Arc Length (Cartesian)
\[L=\int_a^b\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx\]
★ Surface area of revolution: \(S=2\pi\displaystyle\int_a^b f(x)\sqrt{1+(f'(x))^2}\,dx\)
Volume of \(y=\sqrt{x},\ 0\le x\le 4\), revolved about the \(x\)-axis.
Disk method: \(V=\pi\displaystyle\int_0^4 x\,dx=\pi\left[\frac{x^2}{2}\right]_0^4=8\pi\)
✓ \(V=8\pi\)
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