Integration Cheatsheet

Basic Definition

Indefinite Integral: The antiderivative of function $f$ $\int f(x)dx=F(x)+C\text{where }{F}^{\prime }(x)=f(x)$

Definite Integral: Net area under curve from $a$ to $b$ ${\int }_a^{b}f(x)dx=F(b)-F(a)=[F(x){]}_a^b$

Note: For brevity, we write $f$ instead of $f(x)$, $\sin x$ instead of $\sin (x)$, etc. We omit $dx$ when context is clear. Constant of integration $C$ is implied for indefinite integrals.

Basic Integration Rules

Constant and Power Rules

\begin{align} \int c &= cx + C \quad \text{(constant)} \\ \int x^n &= \frac{x^{n+1}}{n+1} + C \quad \text{(power rule, } n \neq -1\text{)} \\ \int \frac{1}{x} &= \ln|x| + C \\ \int \sqrt{x} &= \frac{2}{3}x^{3/2} + C \\ \int \frac{1}{x^2} &= -\frac{1}{x} + C \\ \int \frac{1}{\sqrt{x}} &= 2\sqrt{x} + C \end{align}

Linear Combination Rules

\begin{align} \int cf &= c\int f \quad \text{(constant multiple)} \\ \int [f + g] &= \int f + \int g \quad \text{(sum rule)} \\ \int [f - g] &= \int f - \int g \quad \text{(difference rule)} \end{align}

General Power of Function

$\int [f{]}^n{f}^{\prime }=\frac{[f{]}^{n+1}}{n+1}+C\text{(for }n\ne -1\text{)}$

Special case ($n=-1$): $\int \frac{f^{\prime }}{f}=\ln |f|+C$

Examples: \begin{align} \int \sin^3 x \cos x &= \frac{\sin^4 x}{4} + C \\ \int (x^2+1)^5 \cdot 2x &= \frac{(x^2+1)^6}{6} + C \\ \int \frac{2x}{x^2+1} &= \ln(x^2+1) + C \end{align}

Exponential and Logarithmic Functions

\begin{align} \int e^x &= e^x + C \\ \int a^x &= \frac{a^x}{\ln a} + C \\ \int e^{ax} &= \frac{1}{a}e^{ax} + C \\ \int \frac{1}{x} &= \ln|x| + C \\ \int \ln x &= x\ln x - x + C \\ \int \log_a x &= \frac{x\ln x - x}{\ln a} + C \\ \int \frac{f'}{f} &= \ln|f| + C \end{align}

Trigonometric Functions

\begin{align} \int \sin x &= -\cos x + C \\ \int \cos x &= \sin x + C \\ \int \tan x &= -\ln|\cos x| + C = \ln|\sec x| + C \\ \int \cot x &= \ln|\sin x| + C \\ \int \sec x &= \ln|\sec x + \tan x| + C \\ \int \csc x &= -\ln|\csc x + \cot x| + C \\ \int \sec^2 x &= \tan x + C \\ \int \csc^2 x &= -\cot x + C \\ \int \sec x \tan x &= \sec x + C \\ \int \csc x \cot x &= -\csc x + C \end{align}

Inverse Trigonometric Functions

\begin{align} \int \frac{1}{\sqrt{1-x^2}} &= \arcsin x + C = -\arccos x + C \\ \int \frac{1}{1+x^2} &= \arctan x + C = -\text{arccot } x + C \\ \int \frac{1}{x\sqrt{x^2-1}} &= \text{arcsec } |x| + C \\ \int \frac{1}{\sqrt{a^2-x^2}} &= \arcsin\frac{x}{a} + C \\ \int \frac{1}{a^2+x^2} &= \frac{1}{a}\arctan\frac{x}{a} + C \\ \int \frac{1}{x\sqrt{x^2-a^2}} &= \frac{1}{a}\text{arcsec}\frac{|x|}{a} + C \end{align}

Hyperbolic Functions

\begin{align} \int \sinh x &= \cosh x + C \\ \int \cosh x &= \sinh x + C \\ \int \tanh x &= \ln|\cosh x| + C \\ \int \coth x &= \ln|\sinh x| + C \\ \int \text{sech } x &= \arctan(\sinh x) + C \\ \int \text{csch } x &= \ln\left|\tanh\frac{x}{2}\right| + C \\ \int \text{sech}^2 x &= \tanh x + C \\ \int \text{csch}^2 x &= -\coth x + C \end{align}

Integration Techniques

Substitution (u-substitution)

If $u=g$, then $du=g^{\prime }dx$: $\int f(g)g^{\prime }dx=\int f(u)du$

Example: $\int 2x\cos (x^2)dx=\int \cos udu=\sin u+C=\sin (x^2)+C\text{where }u=x^2$

Integration by Parts

$\int udv=uv-\int vdu$

LIATE rule (priority for choosing $u$): Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential

Common patterns: \begin{align} \int x e^x &= xe^x - e^x + C \\ \int x \sin x &= -x\cos x + \sin x + C \\ \int x^n e^x &= x^n e^x - n\int x^{n-1}e^x \\ \int e^{ax}\sin bx &= \frac{e^{ax}(a\sin bx - b\cos bx)}{a^2+b^2} + C \\ \int e^{ax}\cos bx &= \frac{e^{ax}(a\cos bx + b\sin bx)}{a^2+b^2} + C \end{align}

Partial Fractions

For rational functions $\frac{P}{Q}$ where $\mathrm{deg}P<\mathrm{deg}Q$:

Linear factors: $\frac{A}{x-a}$ $\int \frac{1}{x-a}=\ln |x-a|+C$

Repeated linear factors: $\frac{A}{(x-a{)}^n}$ $\int \frac{1}{(x-a{)}^n}=\frac{(x-a{)}^{-n+1}}{-n+1}+C\text{(for }n\ne 1\text{)}$

Quadratic factors: $\frac{Ax+B}{x^2+bx+c}$

Complete the square and use substitution or arctan formula

Trigonometric Substitution

For $\sqrt{a^2-x^2}$: Use $x=a\sin \theta$, $dx=a\cos \theta ,d\theta$ $\sqrt{a^2-x^2}=a\cos \theta$

For $\sqrt{a^2+x^2}$: Use $x=a\tan \theta$, $dx=a{\sec }^2\theta ,d\theta$ $\sqrt{a^2+x^2}=a\sec \theta$

For $\sqrt{x^2-a^2}$: Use $x=a\sec \theta$, $dx=a\sec \theta \tan \theta ,d\theta$ $\sqrt{x^2-a^2}=a\tan \theta$

Common Integration Patterns

\begin{align} \int e^{ax} &= \frac{1}{a}e^{ax} + C \\ \int \sin ax &= -\frac{1}{a}\cos ax + C \\ \int \cos ax &= \frac{1}{a}\sin ax + C \\ \int (ax+b)^n &= \frac{(ax+b)^{n+1}}{a(n+1)} + C \quad \text{(for } n \neq -1\text{)} \\ \int \frac{1}{ax+b} &= \frac{1}{a}\ln|ax+b| + C \\ \int x^n e^x &= x^n e^x - n\int x^{n-1}e^x \quad \text{(reduction formula)} \end{align}

Special Integrals

\begin{align} \int \frac{1}{x^2+a^2} &= \frac{1}{a}\arctan\frac{x}{a} + C \\ \int \frac{1}{x^2-a^2} &= \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C \\ \int \frac{1}{\sqrt{x^2+a^2}} &= \ln|x+\sqrt{x^2+a^2}| + C \\ \int \frac{1}{\sqrt{x^2-a^2}} &= \ln|x+\sqrt{x^2-a^2}| + C \\ \int \frac{1}{\sqrt{a^2-x^2}} &= \arcsin\frac{x}{a} + C \\ \int \sqrt{a^2-x^2} &= \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\arcsin\frac{x}{a} + C \\ \int \sqrt{x^2+a^2} &= \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\ln|x+\sqrt{x^2+a^2}| + C \end{align}

Properties of Definite Integrals

\begin{align} \int_a^b f\,dx &= -\int_b^a f\,dx \\ \int_a^a f\,dx &= 0 \\ \int_a^b f\,dx &= \int_a^c f\,dx + \int_c^b f\,dx \\ \int_a^b cf\,dx &= c\int_a^b f\,dx \\ \int_a^b [f \pm g]\,dx &= \int_a^b f\,dx \pm \int_a^b g\,dx \end{align}

Symmetry Properties

Even function ($f(-x)=f(x)$): ${\int }_{-a}^{a}fdx=2{\int }_0^{a}fdx$

Odd function ($f(-x)=-f(x)$): ${\int }_{-a}^{a}fdx=0$

Fundamental Theorem of Calculus

Part 1: If $F(x)={\int }_a^{x}f(t),dt$, then $F^{\prime }(x)=f(x)$

Part 2: If $F^{\prime }=f$, then: ${\int }_a^{b}fdx=F(b)-F(a)$

Leibniz Rule (differentiating under integral): $\frac{d}{dx}{\int }_{a(x)}^{b(x)}f(x,t),dt=f(x,b(x))b^{\prime }(x)-f(x,a(x))a^{\prime }(x)+{\int }_{a(x)}^{b(x)}\frac{\partial f}{\partial x}(x,t),dt$

Applications in Statistics

Probability Density Functions

${\int }_{-\infty }^{\infty }f(x)dx=1\text{(normalization)}$

Expected Value

$E[X]={\int }_{-\infty }^{\infty }xf(x)dx$

Variance

$\text{Var}(X)={\int }_{-\infty }^{\infty }(x-\mu {)}^{2}f(x)dx=E[X^2]-(E[X]{)}^2$

Cumulative Distribution Function

$F(x)={\int }_{-\infty }^{x}f(t),dt$

Moment Generating Function

$M_X(t)=E[e^{tX}]={\int }_{-\infty }^{\infty }{e}^{tx}f(x)dx$