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Integration formulas

The antiderivatives and rules every technique question leans on.

$\dfrac{d}{dx}\int_{g(x)}^{h(x)} f(t)\,dt = f(h(x))\,h'(x) - f(g(x))\,g'(x)$
$\int_a^b f(g(x))\,g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du$
$u = g(x)$
$\int u\,dv = uv - \int v\,du$
$u$
$\int\sec x\,dx = \log|\sec x + \tan x| + C$
$\int\tan x\,dx = \log|\sec x| + C$
$\dfrac{d}{dx}\sec x = \sec x\tan x$
$u = \sec x$
$\int\dfrac{dx}{x^2 + a^2} = \dfrac{1}{a}\arctan\dfrac{x}{a} + C$
$Gives the arctan part of an irreducible quadraticA quadratic with no real roots, so it cannot be factored. in partial fractions.$
$\int \dfrac{dx}{\sqrt{a^2 - x^2}} = \arcsin\dfrac{x}{a} + C$
$a > 0$

Trig identities and substitutions

The identities that rewrite leftover powers, and the substitutions that remove roots.

$\sin^2 x = \tfrac12(1 - \cos 2x)$
$\cos^2 x = \tfrac12(1 + \cos 2x)$
$\tan^2 x = \sec^2 x - 1$
$\sin^2 x + \cos^2 x = 1$
$\text{For } \sqrt{a^2 - x^2}$
$x = a\sin\theta$
$a\cos\theta$
$\text{For } \sqrt{a^2 + x^2}$
$x = a\tan\theta$
$a\sec\theta$
$\text{For } \sqrt{x^2 - a^2}$
$x = a\sec\theta$
$a\tan\theta$

Riemann sums and numerical integration

\Delta x = \dfrac{b-a}{n}

$\sum_{i=1}^{n} i = \dfrac{n(n+1)}{2}$
$\sum_{i=1}^{n} i^2 = \dfrac{n(n+1)(2n+1)}{6}$
$M_n = \Delta x\big[f(\bar{x}_1) + \dots + f(\bar{x}_n)\big]$
$\bar{x}_j$
$T_n = \Delta x\left[\tfrac{1}{2}f(x_0) + f(x_1) + \dots + f(x_{n-1}) + \tfrac{1}{2}f(x_n)\right]$
$S_n = \dfrac{\Delta x}{3}\big[f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 4f(x_{n-1}) + f(x_n)\big]$
$n$
$\text{Midpoint rule error}$
$|E_M| \le \dfrac{M(b-a)^3}{24n^2}$
$|f''(x)| \le M$
$\text{Trapezoidal rule error}$
$|E_T| \le \dfrac{M(b-a)^3}{12n^2}$
$|f''(x)| \le M$
$\text{Simpson's rule error}$
$|E_S| \le \dfrac{L(b-a)^5}{180n^4}$
$|f^{(4)}(x)| \le L$

Improper integrals

The integrals you compare against.

$\text{The } p\text{-test at infinity}$
$\int_1^\infty \dfrac{dx}{x^p} \text{ converges} \iff p > 1$
$\frac{1}{p - 1}$
$\text{The } p\text{-test at } 0$
$\int_0^1 \dfrac{dx}{x^p} \text{ converges} \iff p < 1$
$p = 1$
$\int_0^\infty e^{-ax}\,dx = \dfrac{1}{a}$
$a > 0$

Area, volume and average value

Every one is an integral of a slice or a strip.

$\text{Area between curves}$
$\int_a^b (\text{top} - \text{bottom})\,dx$
$\int_c^d (\text{right} - \text{left})\,dy$
$\text{Washers}$
$V = \pi\int_a^b \big(R^2 - r^2\big)\,dx$
$r = 0$
$\text{Known cross-sections}$
$V = \int_a^b A(x)\,dx$
$s^2$
$\bar f \text{ on } [a, b] = \dfrac{1}{b - a}\int_a^b f(x)\,dx$

Work, center of mass, separable equations

The integrals behind the physical applications and the one move for separable equations.

$\text{Hooke's law}$
$F = kx$
$x$
$\text{Weight of a slice of liquid}$
$\rho g\,A(y)\,dy$
$\rho = 1000$
$\bar x\ \text{(rod)} = \dfrac{\int_a^b x\,\rho(x)\,dx}{\int_a^b \rho(x)\,dx}$
$Point masses: the same with sums.$
$\bar x\ \text{(region)} = \dfrac{1}{A}\int_a^b x\,(T - B)\,dx$
$\bar y\ \text{(region)} = \dfrac{1}{2A}\int_a^b \left(T^2 - B^2\right)dx$
$T^2 - B^2$
$\text{Separable: } \dfrac{dy}{dx} = f(x)\,g(y)$
$\int \dfrac{dy}{g(y)} = \int f(x)\,dx + C$
$\text{Solution of } \dfrac{dy}{dt} = ky$
$y = y(0)\,e^{kt}$
$k < 0$

Sequences and series

The limits and tests every sequence and series question starts from.

$\text{Who wins as } n \to \infty$
$\log n \ll n^p \ll a^n \ll n!$
$p > 0$
$\lim_{n \to \infty} r^n = \begin{cases} 0 & |r| < 1 \\ 1 & r = 1 \\ \text{diverges} & \text{otherwise} \end{cases}$
$r = -1$
$\lim_{n \to \infty}\left(1 + \dfrac{a}{n}\right)^n = e^a$
$\sum_{n=0}^{\infty} a r^{n} = \dfrac{a}{1 - r} \quad (|r| < 1)$
$|r| \geq 1$
$\text{The } p\text{-series}$
$\sum_{n=1}^{\infty} \dfrac{1}{n^{p}} \text{ converges} \iff p > 1$
$p = 1$
$\text{Divergence test}$
$\lim_{n\to\infty} a_n \neq 0 \;\Longrightarrow\; \sum a_n \text{ diverges}$
$0$
$\text{Ratio test: } L = \lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|$
$L < 1 \text{ converges},\ L > 1 \text{ diverges}$
$L = 1$
$\text{Alternating series error}$
$|S - S_N| \leq b_{N+1}$
$b_n$

Taylor and Maclaurin series

The general formula, then the six series most power series and Taylor questions build on.

$\text{Taylor series at } x = a$
$\sum_{n=0}^{\infty} \dfrac{f^{(n)}(a)}{n!}(x - a)^{n}$
$a = 0$
$\dfrac{1}{1 - x} = \sum_{n=0}^{\infty} x^{n}$
$|x| < 1$
$e^{x} = \sum_{n=0}^{\infty} \dfrac{x^{n}}{n!}$
$x$
$\sin x = \sum_{n=0}^{\infty} \dfrac{(-1)^{n}x^{2n+1}}{(2n+1)!}$
$x$
$\cos x = \sum_{n=0}^{\infty} \dfrac{(-1)^{n}x^{2n}}{(2n)!}$
$x$
$\log(1 + x) = \sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}x^{n}}{n}$
$-1 < x \leq 1$
$\arctan x = \sum_{n=0}^{\infty} \dfrac{(-1)^{n}x^{2n+1}}{2n+1}$
$-1 \leq x \leq 1$

Must memorize

Integration formulas

Trig identities and substitutions

Riemann sums and numerical integration

Improper integrals

Area, volume and average value

Work, center of mass, separable equations

Sequences and series

Taylor and Maclaurin series

Which topics are you reviewing?

Navigation

Integration formulas

Trig identities and substitutions

Riemann sums and numerical integration

Improper integrals

Area, volume and average value

Work, center of mass, separable equations

Sequences and series

Taylor and Maclaurin series