① Concept Review & Key Formulas to Memorize
Unit 1
Integration by Parts
Used when the integrand is a product of two functions. Choose \(u\) and \(dv\) using LIATE (Logarithmic, Inverse trig, Algebraic, Trig, Exponential).
\(\displaystyle \int u\,dv = uv - \int v\,du\)
- Tabular method speeds up repeated IBP
- For \(\int e^x\sin x\,dx\), apply IBP twice then solve algebraically
Quick Example
\(\displaystyle\int x e^x dx\): Let \(u=x,\,dv=e^x dx\) → \(uv-\int v\,du = xe^x - e^x + C\)
Unit 2
Trigonometric Integrals
Use Pythagorean identities and reduction strategies for \(\int \sin^m x\cos^n x\,dx\).
\(\sin^2x = \tfrac{1-\cos 2x}{2},\quad \cos^2x = \tfrac{1+\cos 2x}{2}\)
\(\sin^2x+\cos^2x=1,\quad 1+\tan^2x=\sec^2x\)
- Odd power: save one factor, convert rest via Pythagorean identity
- Even power: use half-angle identities
Quick Example
\(\displaystyle\int\sin^3 x\,dx = -\cos x + \tfrac{\cos^3 x}{3}+C\)
Unit 3
Trigonometric Substitution
\(\sqrt{a^2-x^2}\): set \(x=a\sin\theta\)
\(\sqrt{a^2+x^2}\): set \(x=a\tan\theta\)
\(\sqrt{x^2-a^2}\): set \(x=a\sec\theta\)
- Always convert back to \(x\) using a reference triangle
- Check domain restrictions for \(\theta\)
Unit 4
Partial Fraction Decomposition
For rational functions where degree of numerator < degree of denominator.
\(\dfrac{1}{(x-a)(x-b)} = \dfrac{A}{x-a}+\dfrac{B}{x-b}\)
\(\dfrac{1}{(x-a)^2} = \dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}\)
- Irreducible quadratic: use \(\dfrac{Ax+B}{x^2+bx+c}\)
- If degree numerator ≥ denominator, do polynomial long division first
Unit 5
Improper Integrals
\(\displaystyle\int_1^\infty \frac{1}{x^p}\,dx\) converges if \(p>1\), diverges if \(p\le 1\)
- Always convert to a limit: \(\displaystyle\lim_{t\to\infty}\int_a^t f(x)\,dx\)
- Comparison Test: if \(0\le f\le g\) and \(\int g\) converges, so does \(\int f\)
Quick Example
\(\displaystyle\int_1^\infty\frac{1}{x^2}dx = \lim_{t\to\infty}\left[-\frac{1}{x}\right]_1^t = 0-(-1)=1\)
Unit 6
Sequences & Limits
\(\displaystyle\lim_{n\to\infty}r^n=0 \text{ if }|r|<1,\quad \lim_{n\to\infty}\frac{n^k}{e^n}=0\)
- Squeeze Theorem applies to sequences
- Monotone Convergence Theorem: bounded + monotone ⟹ converges
- \(\displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = e\)
Unit 7
Series Convergence Tests
Geometric: \(\sum ar^n\) converges iff \(|r|<1\), sum \(=\dfrac{a}{1-r}\)
p-series: \(\sum\frac{1}{n^p}\) converges iff \(p>1\)
Ratio Test: \(L=\lim\left|\dfrac{a_{n+1}}{a_n}\right|\): \(L<1\) conv, \(L>1\) div
Alternating Series: \(\sum(-1)^n b_n\) converges if \(b_n\searrow 0\)
- Divergence Test (first!): if \(\lim a_n\ne 0\), series diverges
- Integral Test: \(\sum a_n\) and \(\int f(x)dx\) same convergence
- Limit Comparison: \(\lim\frac{a_n}{b_n}=L>0\) → same behavior
Unit 8
Power Series & Taylor / Maclaurin Series
\(e^x = \sum_{n=0}^\infty\dfrac{x^n}{n!},\quad \sin x=\sum_{n=0}^\infty\dfrac{(-1)^n x^{2n+1}}{(2n+1)!}\)
\(\cos x=\sum_{n=0}^\infty\dfrac{(-1)^n x^{2n}}{(2n)!},\quad\dfrac{1}{1-x}=\sum_{n=0}^\infty x^n,\;|x|<1\)
- Radius of convergence \(R\) found by Ratio Test
- Interval of convergence: check endpoints separately
- Taylor: \(\displaystyle f(x)=\sum_{n=0}^\infty\frac{f^{(n)}(a)}{n!}(x-a)^n\)
Unit 9
Parametric Equations
\(\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt},\quad\dfrac{d^2y}{dx^2}=\dfrac{d(dy/dx)/dt}{dx/dt}\)
Arc length: \(\displaystyle L=\int_\alpha^\beta\sqrt{\left(\tfrac{dx}{dt}\right)^2+\left(\tfrac{dy}{dt}\right)^2}\,dt\)
Unit 10
Polar Coordinates
Area: \(\displaystyle A=\frac{1}{2}\int_\alpha^\beta r^2\,d\theta\)
Arc length: \(\displaystyle L=\int_\alpha^\beta\sqrt{r^2+\left(\tfrac{dr}{d\theta}\right)^2}\,d\theta\)
- \(x=r\cos\theta,\;y=r\sin\theta,\;r^2=x^2+y^2\)
- Cardioid \(r=a(1+\cos\theta)\), Lemniscate \(r^2=a^2\cos 2\theta\)