Calculus II

UNIT 1

Integration Techniques

Master integration by parts, trigonometric substitution, and partial fractions — the three pillars of Calculus II integration.

★ Memorize

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

\(\displaystyle\int \frac{dx}{x^2+a^2} = \frac{1}{a}\arctan\!\frac{x}{a}+C\)

Trig sub: \(x=a\sin\theta\) when \(\sqrt{a^2-x^2}\) appears; \(x=a\tan\theta\) when \(\sqrt{a^2+x^2}\) appears.

Quick Example

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

UNIT 2

Improper Integrals

Integrals over infinite intervals or with singularities — evaluated as limits.

★ Memorize

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

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

Quick Example

\(\displaystyle\int_1^{\infty}\frac{dx}{x^2} = \lim_{t\to\infty}\left[-\frac{1}{x}\right]_1^t = 0-(-1) = 1\). Converges.

UNIT 3

Sequences & Series

Convergence tests for infinite series are the heart of Calculus II.

★ Memorize Convergence Tests

Geometric: \(\displaystyle\sum_{n=0}^{\infty}ar^n = \frac{a}{1-r}\) if \(|r|<1\)

Ratio Test: \(L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\). Converges if \(L<1\), diverges if \(L>1\)

p-series: \(\displaystyle\sum \frac{1}{n^p}\) converges if \(p>1\)

Quick Example

\(\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2}\) converges (\(p=2>1\)). Sum \(= \frac{\pi^2}{6}\).

UNIT 4

Power Series & Taylor/Maclaurin Series

A function expressed as an infinite polynomial; the Taylor series centered at \(a\).

★ Key Maclaurin Series

\(e^x = \displaystyle\sum_{n=0}^{\infty}\frac{x^n}{n!},\quad |x|<\infty\)

\(\sin x = \displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!},\quad\cos x = \displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}\)

\(\frac{1}{1-x}=\displaystyle\sum_{n=0}^{\infty}x^n,\quad |x|<1\)

Quick Example

Find the Maclaurin series for \(e^{-x^2}\):
Replace \(x\) with \(-x^2\) in the series for \(e^x\):
\(\displaystyle e^{-x^2}=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{n!}\)

UNIT 5

Parametric Curves & Polar Coordinates

Curves defined by \(x=f(t),\,y=g(t)\), or in polar form \(r=f(\theta)\).

★ Key Formulas

\(\displaystyle\frac{dy}{dx}=\frac{dy/dt}{dx/dt},\quad \text{Arc length} = \int_a^b\!\sqrt{\!\left(\frac{dx}{dt}\right)^{\!2}+\left(\frac{dy}{dt}\right)^{\!2}}\,dt\)

Polar area: \(\displaystyle A=\frac{1}{2}\int_\alpha^\beta r^2\,d\theta\)

Quick Example

For the circle \(r=2\), area \(=\frac{1}{2}\displaystyle\int_0^{2\pi}4\,d\theta=\frac{1}{2}\cdot 4\cdot 2\pi = 4\pi\). ✓

Core Concepts & Formulas to Memorize

Full Answer Review & Explanations

Answer Key & Explanations