Calculus II

1

Integration by Parts

IBP

\(\int u\,dv = uv - \int v\,du\)

🧠 Memorize: LIATE Rule

Logarithmic · Inverse trig · Algebraic · Trig · Exponential — choose \(u\) from left to right.

When integrand is a product of two different function types, use IBP.

Example

\(\int x e^x dx\): Let \(u=x\), \(dv=e^x dx\). Then \(du=dx\), \(v=e^x\).
Answer: \(xe^x - e^x + C = e^x(x-1)+C\)

2

Trigonometric Integrals

Trig Int

\(\sin^2 x = \tfrac{1-\cos 2x}{2}\quad \cos^2 x = \tfrac{1+\cos 2x}{2}\)

\(\int \sin^m x\cos^n x\,dx\): if \(m\) odd, factor out \(\sin x\) and substitute \(u=\cos x\). If \(n\) odd, substitute \(u=\sin x\).

If both powers even, use the half-angle identities above.

Example

\(\int \sin^2 x\,dx = \int\tfrac{1-\cos 2x}{2}\,dx = \tfrac{x}{2} - \tfrac{\sin 2x}{4} + C\)

3

Trigonometric Substitution

Trig Sub

\(\sqrt{a^2-x^2}\Rightarrow x=a\sin\theta\)
\(\sqrt{a^2+x^2}\Rightarrow x=a\tan\theta\)
\(\sqrt{x^2-a^2}\Rightarrow x=a\sec\theta\)

🧠 Memorize

sin for minus inside square root, tan for plus, sec for minus outside.

4

Partial Fraction Decomposition

PFD

\(\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a}+\frac{B}{x-b}\)

Degree of numerator must be less than degree of denominator. If not, do polynomial long division first.

Repeated factor \((x-a)^2\): decompose as \(\frac{A}{x-a}+\frac{B}{(x-a)^2}\).

5

Improper Integrals

Improper

\(\int_a^{\infty} f(x)\,dx = \lim_{t\to\infty}\int_a^t f(x)\,dx\)

\(\int_1^{\infty} \frac{1}{x^p}\,dx\) converges if \(p>1\), diverges if \(p\le 1\).

Example

\(\int_1^{\infty}\frac{1}{x^2}\,dx = \left[-\frac{1}{x}\right]_1^{\infty} = 0-(-1)=1\) ✓ Converges

6

Sequences & Convergence

Sequences

\(\lim_{n\to\infty} a_n = L \Rightarrow \text{converges to }L\)

L'Hôpital's Rule applies when sequence limit gives \(\frac{\infty}{\infty}\) or \(\frac{0}{0}\) form.

Squeeze Theorem: if \(b_n\le a_n\le c_n\) and \(b_n,c_n\to L\), then \(a_n\to L\).

7

Geometric & p-Series

Series

Geometric: \(\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r},\;|r|<1\)
p-Series: \(\sum_{n=1}^{\infty}\frac{1}{n^p}\) converges iff \(p>1\)

🧠 Key Values

Harmonic series \(\sum 1/n\) diverges (\(p=1\)). Basel problem \(\sum 1/n^2 = \pi^2/6\) converges (\(p=2\)).

8

Convergence Tests

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: \(\sum(-1)^n b_n\) converges if \(b_n\searrow 0\)

Comparison Test: \(0\le a_n\le b_n\). If \(\sum b_n\) converges → \(\sum a_n\) converges. If \(\sum a_n\) diverges → \(\sum b_n\) diverges.

Divergence Test (necessary): if \(\lim a_n\ne 0\), series diverges.

9

Power Series & Radius of Convergence

Power

\(\sum_{n=0}^{\infty} c_n(x-a)^n,\quad R = \lim_{n\to\infty}\left|\frac{c_n}{c_{n+1}}\right|\)

Interval of convergence: \((a-R, a+R)\). Always check endpoints separately.

10

Taylor & Maclaurin Series

Taylor

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

🧠 Must Memorize

\(e^x = \sum\frac{x^n}{n!}\)
\(\sin x = \sum\frac{(-1)^n x^{2n+1}}{(2n+1)!}\)
\(\cos x = \sum\frac{(-1)^n x^{2n}}{(2n)!}\)
\(\frac{1}{1-x} = \sum x^n,\;|x|<1\)

11

Parametric Equations

Parametric

\(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\qquad \frac{d^2y}{dx^2}=\frac{d(dy/dx)/dt}{dx/dt}\)

Arc length: \(L=\int_\alpha^\beta\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt\)

12

Polar Coordinates

Polar

Area: \(A=\frac{1}{2}\int_\alpha^\beta r^2\,d\theta\)
Arc Length: \(L=\int_\alpha^\beta\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta\)

Conversion: \(x=r\cos\theta,\ y=r\sin\theta,\ r^2=x^2+y^2\).

13

Volumes of Solids of Revolution

Volumes

Disk: \(V=\pi\int_a^b [f(x)]^2\,dx\)
Washer: \(V=\pi\int_a^b [R(x)^2-r(x)^2]\,dx\)
Shell: \(V=2\pi\int_a^b x\,f(x)\,dx\)

Disk/Washer: rotating about x-axis, integrate \(dx\). Shell: rotating about y-axis, integrate \(dx\).

14

Separable Differential Equations

ODE

\(\frac{dy}{dx}=g(x)h(y)\Rightarrow \frac{dy}{h(y)}=g(x)\,dx\)

Separate variables, integrate both sides, solve for \(y\) if possible.

Example

\(\frac{dy}{dx}=2xy\Rightarrow \frac{dy}{y}=2x\,dx\Rightarrow \ln|y|=x^2+C\Rightarrow y=Ae^{x^2}\)

Core Concepts

Answer Key & Explanations