Calculus 2 Memorization Sheet

Quick Reference

Technique

Integration by Parts $\int u\,dv = uv – \int v\,du$

Area & Average Value

Area under the curve $A = \displaystyle\int_a^b f(x)\,dx$

Average value $f_{\text{avg}} = \dfrac{1}{b-a}\displaystyle\int_a^b f(x)\,dx$

Volumes of Revolution

Shell method (y-axis) $V = \displaystyle\int_a^b 2\pi x\, f(x)\, dx$

Arc Length

Arc length formula $L = \displaystyle\int_a^b \sqrt{1 + \bigl(f'(x)\bigr)^2}\, dx$

Series

Taylor series $f(x)=\sum\dfrac{f^{(n)}(a)}{n!}(x-a)^n$

Maclaurin series $f(x)=\sum\dfrac{f^{(n)}(0)}{n!}x^n$

Derivatives

Derivative of arctan $\dfrac{d}{dx}(\arctan x) = \dfrac{1}{1+x^{2}}$

Derivative of tan $\dfrac{d}{dx}(\tan x) = \sec^{2}x$

Derivative of sec $\dfrac{d}{dx}(\sec x) = \sec x\tan x$

Integrals

Integral of arctan form $\displaystyle\int\dfrac{1}{1+x^{2}}\,dx = \arctan x + C$

Integral of sec² $\displaystyle\int\sec^{2}x\,dx = \tan x + C$

Integral of sec $\displaystyle\int\sec\theta\,d\theta = \ln|\sec\theta + \tan\theta| + C$

Integral of sec tan $\displaystyle\int\sec x\tan x\,dx = \sec x + C$

Identities

Pythagorean $\sin^{2}x + \cos^{2}x = 1$

$1 + \tan^{2}x = \sec^{2}x$

$1 + \cot^{2}x = \csc^{2}x$

Power-reduction $\cos^{2}\theta = \dfrac{1+\cos 2\theta}{2}$

$\sin^{2}\theta = \dfrac{1-\cos 2\theta}{2}$

Double-angle $\sin 2\theta = 2\sin\theta\cos\theta$

Difference of cubes $a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})$

Sum of cubes $a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})$

Integration by Parts

$$\int u\,dv = uv – \int v\,du$$

How to choose $u$ (LIATE)

Logarithmic → Inverse trig → Algebraic → Trig → Exponential

Pick $u$ as the first type that appears. The rest becomes $dv$.

Area Under a Curve & Average Value

Area under the curve

$$A = \int_a^b f(x)\,dx$$

Average value of a function

$$f_{\text{avg}} = \dfrac{1}{b-a}\int_a^b f(x)\,dx$$

Remember

The average value is the height of a rectangle that has the same base $b-a$ and the same area as the region under the curve.

Shell Method (about the y-axis)

$$V = \int_a^b 2\pi x\, f(x)\, dx$$

When to use

Revolve a region described by $ y = f(x) $ around the y-axis. Radius of each shell = $ x $, height = $ f(x) $, thickness = $ dx $.

Arc Length

$$L = \int_a^b \sqrt{1 + \bigl(f'(x)\bigr)^2}\, dx$$

Remember

First find $ f'(x) $, square it, add 1, take the square root, then integrate from $ a $ to $ b $.

Taylor Series

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

Remember

The series is centered at $a$. Compute successive derivatives, evaluate them at $a$, and form the coefficients $\frac{f^{(n)}(a)}{n!}$.

Maclaurin Series

$$f(x) = \sum_{n=0}^{\infty} \dfrac{f^{(n)}(0)}{n!}x^n$$

Remember

A Maclaurin series is simply a Taylor series centered at $a = 0$. It is the most common and useful form.

Maclaurin Series to Memorize

$e^x$

$$\sum_{n=0}^{\infty} \dfrac{x^n}{n!} \qquad \text{(all } x\text{)}$$

$\sin x$

$$\sum_{n=0}^{\infty} \dfrac{(-1)^n}{(2n+1)!}x^{2n+1} \qquad \text{(all } x\text{)}$$

$\cos x$

$$\sum_{n=0}^{\infty} \dfrac{(-1)^n}{(2n)!}x^{2n} \qquad \text{(all } x\text{)}$$

$\dfrac{1}{1-x}$

$$\sum_{n=0}^{\infty} x^n \qquad (|x|<1)$$

$\ln(1+x)$

$$\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{n}x^n \qquad (-1 < x \le 1)$$

Tip

These five series are the building blocks. Almost every other Maclaurin series problem is solved by substituting into one of these.

Derivative of $\arctan x$

$$\frac{d}{dx}\bigl(\arctan x\bigr) = \frac{1}{1+x^{2}}$$

Also useful

$\displaystyle\int \frac{1}{1+x^{2}}\,dx = \arctan x + C$

Derivative of $\tan x$

$$\frac{d}{dx}\bigl(\tan x\bigr) = \sec^{2}x$$

Also useful

$\displaystyle\int \sec^{2}x\,dx = \tan x + C$

Derivative of $\sec x$

$$\frac{d}{dx}\bigl(\sec x\bigr) = \sec x\tan x$$

Also useful

$\displaystyle\int \sec x\tan x\,dx = \sec x + C$

Integral of $\sec\theta$

$$\int\sec\theta\,d\theta = \ln|\sec\theta + \tan\theta| + C$$

Sum & Difference of Cubes

$a^{3} – b^{3} = (a-b)(a^{2} + ab + b^{2})$

$a^{3} + b^{3} = (a+b)(a^{2} – ab + b^{2})$

Pythagorean Identities

$\sin^{2}x + \cos^{2}x = 1$

$1 + \tan^{2}x = \sec^{2}x$

$1 + \cot^{2}x = \csc^{2}x$

Power-Reduction & Double-Angle

$\cos^{2}\theta = \dfrac{1+\cos 2\theta}{2}$

$\sin^{2}\theta = \dfrac{1-\cos 2\theta}{2}$

$\sin 2\theta = 2\sin\theta\cos\theta$

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