Integrals#
Indefinite integrals#
Description |
Equations |
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Indefinite integral (antiderivative) |
\(F(x) = \displaystyle\int f(x) \ dx \newline F'(x) = f(x)\) |
Antiderivative as a family of functions |
If \(F\) is an antiderivative of \(f\), \(C\) is a constant, |
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 |
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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 |
\(\int_a^b f(x) \ dx = \lim\limits_{n\to\infty} \sum\limits_{i=1}^n f(x_i)\Delta x\) |
Properties of definite integrals#
Description |
Equations |
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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 |
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Fundamental Theorem of Calculus I |
\(g(x) = \displaystyle\int_a^x f(t) \, dt\) |
Fundamental Theorem of Calculus II |
\(\displaystyle\int_a^b f(x) \, dx = F(b) - F(a)\) |
Net Change Theorem |
\(\displaystyle\int_a^b F'(x) \, dx = F(b) - F(a)\) |
Substitution rule#
Description |
Equations |
---|---|
Substitution rule (u-substitution) |
\(\displaystyle\int f(g(x)) g'(x) \ dx = \int f(u) \ du\) |
Substitution rule for definite integrals |
\(\displaystyle\int_a^b f(g(x))g'(x) \ dx = \int_{g(a)}^{g(b)} f(u) \ du\) |
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 |
\(\sin A \cos B = \frac{1}{2}[\sin(A-B) + \sin(A+B)]\) |
Trig identity for solving |
\(\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]\) |
Trig identity for solving |
\(\cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]\) |