Integrals#

Indefinite 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 \(F(x) + C\)

Table of indefinite integrals#

Function \(f(x)\)

Antiderivative \(F(x)\)

Function \(f(x)\)

Antiderivative \(F(x)\)

\(x^n\)

\(\dfrac{x^{n+1}}{n+1} + C\)

\(\dfrac{1}{x}\)

\(\ln\lvert x \rvert + C\)

\(e^x\)

\(e^x + C\)

\(b^x\)

\(\dfrac{b^x}{\ln b} + C\)

\(\sin x\)

\(-\cos x + C\)

\(\cos x\)

\(\sin x + C\)

\(\dfrac{1}{x^2 + a^2}\)

\(\dfrac{1}{a}\arctan \left(\dfrac{x}{a}\right) + C\)

\(\dfrac{1}{\sqrt{a^2-x^2}}\)

\(\arcsin \left(\dfrac{x}{a}\right) + C\)

Definite integrals as Riemann sums#

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

\(\int_a^b f(x) \ dx = \lim\limits_{n\to\infty} \sum\limits_{i=1}^n f(x_i)\Delta x\)
\(\Delta x = \frac{b-a}{n} \newline x_i = a+i\Delta x\)

Properties of definite integrals#

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

\(\int_a^b cf(x) \ dx = c\int_a^b f(x) \ dx\)

Fundamental theorem of calculus#

Description

Equations

Fundamental Theorem of Calculus I
(If \(f\) is continuous on \([a,b]\))

\(g(x) = \displaystyle\int_a^x f(t) \, dt\)
\(g'(x) = f(x)\)

Fundamental Theorem of Calculus II
(If \(f\) 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)

\(\displaystyle\int_a^b F'(x) \, dx = F(b) - F(a)\)

Substitution rule#

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

\(\int_{-a}^a f(x) \ dx = 0\)

Techniques of Integration#

Integration by parts#

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

\(\int_a^b fg' \ dx = [fg]_a^b - \int_a^b f'g \ dx\)

Trigonometric integrals#

Description

Equations

Integral of even power of sine and cosine using 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
\(\int \cos(mx)\cos(nx) \, dx\)

\(\cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]\)