Integrals

Integrals#

Description	Equations
Indefinite integral (antiderivative)	\(F(x) = \displaystyle\int f(x) \ dx \newline F'(x) = f(x)\)
Antiderivative as a family of functions (Plus \(C\)!)	If \(F\) is an antiderivative of \(f\), \(C\) is a constant, then the most general antiderivative is \(F(x) + C\)

Function \(f(x)\)	Antiderivative \(F(x)\)	Function \(f(x)\)	Antiderivative \(F(x)\)
\(x^n\)	\(\dfrac{x^{n+1}}{n+1}+C\)	\(\dfrac{1}{x}\)	\(\ln\lvert x \rvert + C\)
\(e^x\)	\(e^x + C\)	\(b^x\)	\(\dfrac{b^x}{\ln b} + C\)
\(\sin x\)	\(-\cos x + C\)	\(\cos x\)	\(\sin x + C\)
\(\sec^2 x\)	\(\tan x + C\)	\(csc^2 x\)	\(-\cot x + C\)
\(\sec x\tan x\)	\(\sec x + C\)	\(\csc x\cot x\)	\(-\csc x + C\)
\(\dfrac{1}{x^2 + a^2}\)	\(\dfrac{1}{a}\arctan \left(\dfrac{x}{a}\right) + C\)	\(\dfrac{1}{\sqrt{a^2-x^2}}\)	\(\arcsin \left(\dfrac{x}{a}\right) + C\)

Description	Equations
Area	\(A = \lim\limits_{n\to\infty} R_{n} = \lim\limits_{n\to\infty} \sum\limits_{i=1}^{n}f(x_i)\Delta x\)
Definite integral	\(\int_{a}^{b} f(x) \ dx = \lim\limits_{n\to\infty} \sum\limits_{i=1}^{n}f(x_i^*)\Delta x\)
Operational definition of definite integral as Riemann sum	\(\int_a^b f(x) \ dx = \lim\limits_{n\to\infty} \sum\limits_{i=1}^n f(x_i)\Delta x\) \(\Delta x = \frac{b-a}{n} \newline x_i = a+i\Delta x\)

Description	Equations
Reversing the bounds changes the sign of definite integrals	\(\int_a^b f(x) \ dx = -\int_b^a f(x) \ dx\)
Definite integral is zero if upper and lower bounds are the same	\(\int_a^a f(x) \ dx = 0\)
Definite integrals of constant	\(\int_a^b c \ dx = c(b-a)\)
Addition and subtraction of definite integrals	\(\int_a^b [f(x) \pm g(x)] \ dx \newline = \int_a^b f(x) \ dx \pm \int_a^b g(x) \ dx\)
Constant multiple of definite integrals	\(\int_a^b cf(x) \ dx = c\int_a^b f(x) \ dx\)

Description	Equations
Fundamental theorem of calculus I (\(f\) is continuous on \([a,b]\))	\(g(x) = \displaystyle\int_a^x f(t) \ dt \newline g'(x) = f(x)\)
Fundamental theorem of calculus II (\(f\) is continuous on \([a,b]\))	\(\displaystyle\int_a^b f(x) \ dx = F(b) - F(a)\) where \(F\) is any antiderivative of \(f\)
Net change theorem The integral of a rate of change is the net change	\(\displaystyle\int_a^b F'(x) \ dx = F(b) - F(a)\)

Description	Equations
Substitution rule (u-substitution) \(u \equiv g(x)\)	\(\displaystyle\int f(g(x)) g'(x) \ dx = \int f(u) \ du\)
Substitution rule for definite integrals \(u \equiv g(x)\)	\(\displaystyle\int_a^b f(g(x))g'(x) \ dx = \int_{g(a)}^{g(b)} f(u) \ du\)
Integrals of even functions	\(\int_{-a}^a f(x) \ dx = 2 \int_{0}^a f(x) \ dx\)
Integrals of odd functions	\(\int_{-a}^a f(x) \ dx = 0\)

Description	Equations
Integration by parts	\(\int f(x)g'(x) \ dx \newline = f(x)g(x) - \int g(x)f'(x) \ dx\)
Integration by parts	\(\int u \ dv = uv - \int v \ du\)
Integration by parts for definite integrals	\(\int_a^b fg' \ dx = [fg]_a^b - \int_a^b f'g \ dx\)

Description	Equations
Integral of odd power of cosine \((u = \sin x)\)	\(\int \sin^m(x)\cos^{2k+1}(x) \ dx \newline = \int \sin^m(x) [\cos^2 (x)]^k \ dx \newline = \int \sin^m(x)[1-\sin^2(x)]^k \ dx\)
Integral of odd power of sine \((u = \cos x)\)	\(\int \sin^{2k+1}(x)\cos^{n}(x) \ dx \newline = \int [\sin^2 (x)]^k \cos^n(x) \sin(x) \ dx \newline = \int [1-\cos^2(x)]^k \cos^n(x) \sin(x) \ dx\)
Integral of even power of sine and cosine use trig identities	\(\sin^2(x) = \frac{1}{2}(1-\cos(2x)) \newline \cos^2(x) = \frac{1}{2}(1+\cos(2x)) \newline \sin(x)\cos(x) = \frac{1}{2}\sin(2x)\)
Trig identity for solving \(\int \sin(mx)\cos(nx) \ dx\)	\(\sin A \cos B = \frac{1}{2}[\sin(A-B) + \sin(A+B)]\)
Trig identity for solving \(\int \sin(mx)\sin(nx) \ dx\)	\(\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]\)
Trig identity for solving \(\int \cos(mx)\cos(nx) \ dx\)	\(\cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]\)