§1 · Integration Techniques
Evaluate \(\displaystyle\int x^2 e^x\,dx\).
LIATE Rule: choose \(u\) in order — Log · Inverse trig · Algebraic · Trig · Exponential. Pick the one that SHRINKS when differentiated.
Worked Pattern
\(\int x e^x dx\): set \(u=x,\;dv=e^x dx\) → \(xe^x - e^x + C\). For \(x^2\), apply IBP twice.
Evaluate \(\displaystyle\int \frac{dx}{\sqrt{9-x^2}}\).
TRIG-SUB TABLE:
\(\sqrt{a^2-x^2}\) → \(x=a\sin\theta\) | \(\sqrt{a^2+x^2}\) → \(x=a\tan\theta\) | \(\sqrt{x^2-a^2}\) → \(x=a\sec\theta\)
\(\sqrt{a^2-x^2}\) → \(x=a\sin\theta\) | \(\sqrt{a^2+x^2}\) → \(x=a\tan\theta\) | \(\sqrt{x^2-a^2}\) → \(x=a\sec\theta\)
Decompose and integrate \(\displaystyle\int \frac{2x+3}{(x-1)(x+2)}\,dx\).
COVER-UP trick: for \(\frac{N(x)}{(x-a)(x-b)}\), cover \((x-a)\) and plug \(x=a\) to find numerator of \(\frac{A}{x-a}\) instantly.
Setup
\(\dfrac{2x+3}{(x-1)(x+2)} = \dfrac{A}{x-1}+\dfrac{B}{x+2}\). Multiply both sides by \((x-1)(x+2)\), then set \(x=1\) and \(x=-2\).
Evaluate \(\displaystyle\int \sin^3 x\cos^2 x\,dx\).
ODD-POWER trick: if \(\sin^m x\cos^n x\) has an ODD power, peel one factor for \(du\) and convert the rest with \(\sin^2=1-\cos^2\) (or vice versa).
§2 · Improper Integrals & Applications
Does \(\displaystyle\int_1^{\infty}\frac{1}{x^p}\,dx\) converge? If so, when?
p-INTEGRAL rule: \(\int_1^\infty \frac{dx}{x^p}\) converges iff \(p>1\). Equals \(\frac{1}{p-1}\). Flip: \(\int_0^1 \frac{dx}{x^p}\) converges iff \(p<1\).
The arc length formula for \(y=f(x)\) from \(a\) to \(b\) is:
PYTHAGORAS in disguise: think tiny right triangle: \(ds = \sqrt{(dx)^2+(dy)^2} = \sqrt{1+(y')^2}\,dx\). Arc length = sum of all \(ds\).
Region bounded by \(y=x^2\) and \(y=x\) is revolved around the \(x\)-axis. Volume \(=\)?
WASHER = OUTER − INNER: \(V = \pi\int_a^b\!\bigl[R(x)^2 - r(x)^2\bigr]dx\). Never forget to square each radius separately — NOT \((R-r)^2\)!
§3 · Sequences & Series
Find the sum: \(\displaystyle\sum_{n=0}^{\infty} \frac{3}{4^n}\).
GEO-SUM: \(\sum_{n=0}^\infty ar^n = \dfrac{a}{1-r}\) when \(|r|<1\). Identify \(a\) = first term, \(r\) = ratio. If \(|r|\geq 1\) → DIVERGE.
Apply the Ratio Test to \(\displaystyle\sum_{n=1}^{\infty}\frac{n!}{n^n}\). What is the result?
RATIO TEST shortcut: compute \(L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\). If \(L<1\) → converge · \(L>1\) → diverge · \(L=1\) → inconclusive. Useful with factorials & exponentials.
Key Limit Needed
Recall: \(\lim_{n\to\infty}\left(1+\tfrac{1}{n}\right)^n = e\), so \(\lim_{n\to\infty}\left(\tfrac{n}{n+1}\right)^n = \tfrac{1}{e}\).
Which statement about \(\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}\) is TRUE?
AST 3-check: (1) alternating signs? (2) \(b_n > 0\)? (3) \(b_n\) decreasing? (4) \(b_n \to 0\)? If all YES → converges. But "converges" ≠ "absolutely converges"!
Does \(\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2+\sin n}\) converge?
LCT (Limit Comparison): compare with a known series \(\sum b_n\). If \(\lim_{n\to\infty}\frac{a_n}{b_n} = L\) (finite, positive) → same fate. Ignore lower-order noise in denominator.
§4 · Power Series & Taylor Series
Find the radius of convergence of \(\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n x^n}{n+1}\).
RATIO for ROC: \(R = \lim_{n\to\infty}\left|\frac{a_n}{a_{n+1}}\right|\). Once you have \(R\), check endpoints separately — convergence there is never automatic!
The Maclaurin series for \(e^x\) is used to find \(\displaystyle\int_0^1 e^{-x^2}\,dx\) as a series. What is the first three terms?
SUBSTITUTE trick: replace \(x\) with \(-x^2\) in \(e^x = \sum\frac{x^n}{n!}\), then integrate term-by-term. Works whenever direct integration is impossible.
Step 1
\(e^{-x^2} = 1 - x^2 + \dfrac{x^4}{2!} - \dfrac{x^6}{3!} + \cdots\). Then integrate from 0 to 1.
The Taylor remainder \(R_n(x)\) is bounded by Lagrange's formula. Which is it?
LAGRANGE BOUND: \(|R_n(x)| \leq \dfrac{M\,|x-a|^{n+1}}{(n+1)!}\), where \(M = \max|f^{(n+1)}|\). USE this to find how many terms give desired accuracy.
§5 · Differential Equations & Parametric/Polar
Solve \(\dfrac{dy}{dx} = xy\), with \(y(0)=2\).
SEPARATE-INTEGRATE: move all \(y\) to one side, \(x\) to other → \(\frac{dy}{y}=x\,dx\) → integrate both → \(\ln|y|=\frac{x^2}{2}+C\) → exponentiate → apply IC.
For \(x=t^2,\; y=t^3\), find \(\dfrac{d^2y}{dx^2}\) in terms of \(t\).
SECOND DERIV parametric: \(\dfrac{d^2y}{dx^2} = \dfrac{\frac{d}{dt}\!\left(\frac{dy/dt}{dx/dt}\right)}{dx/dt}\). Don't differentiate \(\frac{dy}{dx}\) with respect to \(t\) and forget to divide by \(\frac{dx}{dt}\) again!
Area enclosed by \(r = 2\cos\theta\):
POLAR AREA: \(A = \frac{1}{2}\int_\alpha^\beta r^2\,d\theta\). The circle \(r=2\cos\theta\) completes one full loop as \(\theta\) goes from \(-\pi/2\) to \(\pi/2\). Don't use \(0\) to \(2\pi\) (double counts).
Which of the following series DIVERGES?
DIVERGENCE TEST first: if \(\lim_{n\to\infty}a_n \neq 0\), the series DIVERGES — no further work needed. But if \(a_n\to 0\), you still can't conclude convergence (e.g. harmonic series).
Evaluate \(\displaystyle\int e^x \sin x\,dx\).
CYCLIC IBP: apply IBP twice and the original integral reappears on the right. Move it to the left and solve algebraically. Don't panic — it's supposed to happen!
Key Insight
Let \(I = \int e^x\sin x\,dx\). After two IBP steps you get \(I = e^x\sin x - e^x\cos x - I\), so \(2I = e^x(\sin x - \cos x)\).
BOSS QUESTION: Which of the following is TRUE?
CONVERGENCE MASTER TABLE: Geometric: \(|r|<1\) → conv. | p-series: \(p>1\) → conv. | Harmonic \(\sum\frac{1}{n}\): DIVERGES. | Alternating harmonic: converges conditionally. | \(\sum\frac{1}{n\ln n}\): diverges (integral test).