Topic 01 · Integration Techniques
Integration by Parts & Substitution
\(\int u\,dv = uv - \int v\,du\)
\(\int f(g(x))g'(x)\,dx = \int f(u)\,du,\quad u=g(x)\)
LIATE rule for IBP: choose \(u\) as Logarithm, Inverse trig, Algebraic, Trig, Exponential (in that priority order).
Topic 02 · Partial Fractions & Trig Sub
Rational Functions & Trig Substitution
For \(\sqrt{a^2-x^2}\): let \(x=a\sin\theta\)
For \(\sqrt{a^2+x^2}\): let \(x=a\tan\theta\)
For \(\sqrt{x^2-a^2}\): let \(x=a\sec\theta\)
Partial fractions decompose rational functions into simpler pieces before integrating.
Topic 03 · Improper Integrals
Convergence & Divergence
\(\int_1^{\infty}\frac{1}{x^p}\,dx\) converges iff \(p>1\)
Replace infinite limit with \(t\), integrate, then take \(\lim_{t\to\infty}\). Use comparison test when direct integration is difficult.
Topic 04 · Sequences & Series
Convergence Tests
Ratio Test: \(L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\)
\(L<1\): converge · \(L>1\): diverge · \(L=1\): inconclusive
Alternating Series: converges if \(b_n\) decreasing \(\to 0\)
Topic 05 · Power Series
Taylor & Maclaurin Series
\(e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}\)
\(\sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}\)
\(\cos x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}\)
\(\frac{1}{1-x} = \sum_{n=0}^{\infty}x^n,\;|x|<1\)
Topic 06 · Polar Coordinates & Arc Length
Area & Length in Polar Form
Area: \(A=\frac{1}{2}\int_\alpha^\beta r^2\,d\theta\)
Arc Length: \(L=\int\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt\)
📝 Worked Example
Evaluate \(\int x e^x dx\).
Let \(u=x,\;dv=e^x dx\) (LIATE: Algebraic before Exponential)
Then \(du=dx,\;v=e^x\)
Answer: \(xe^x - e^x + C = e^x(x-1)+C\)