Integrals

Abstract

After reading this chapter you know:

• what an integral is,

• what definite and indefinite integrals are,

• what an anti-derivative is and how it is related to the indefinite integral,

• what the area under a curve is and how it is related to the definite integral,

• how to solve some integrals and

• how integrals can be applied, with specific examples in convolution and the calculation of expected value.

Appendices

Symbols Used in This Chapter (in Order of Their Appearance)

∫.	Indefinite integral
$\underset{a}{\overset{b}{\int \limits }}.$	Definite integral between the limits a and b
arctan	Inverse of tangent function
arcsin	Inverse of sine function
≈	Approximately equal to

Overview of Equations for Easy Reference

Indefinite integral

f(x) =  ∫ f?>′(x)dx

Basic indefinite integrals

$$ {\displaystyle \begin{array}{ll}f(x)=A\ \left(\mathrm{constant}\right),\hfill & \int f(x) dx= Ax+C\hfill \\ {}f(x)={x}^n\kern0.5em ,\kern0.75em n\in \mathrm{\mathbb{C}}\wedge \kern0.5em \mathrm{n}\ne -1,\hfill & \int f(x) dx=\frac{x^{n+1}}{n+1}+C\hfill \\ {}f(x)={e}^{ax}, \kern0.5em a\in \mathrm{\mathbb{C}}\wedge \kern0.5em \mathrm{a}\ne 0\hfill & \int f(x) dx=\frac{1}{a}{e}^{ax}+C\hfill \\ {}f(x)=\frac{1}{x}\kern0.5em \left(\mathrm{or}\ {x}^{-1}\right)\hfill & \int f(x) dx=\left\{\begin{array}{c}\hfill lnx+\mathrm{C}\kern3.25em \mathrm{if}\ x>0\hfill \\ {}\hfill \mathit{\ln}\left(-x\right)+\mathrm{C}\kern1.25em \mathrm{if}\ x<0\hfill \end{array}\right.\hfill \\ {}f(x)=\sin ax,\kern0.75em a\in \mathrm{\mathbb{C}}\wedge \mathrm{a}\ne 0,\hfill & \int f(x) dx=-\frac{1}{a}\cos ax+C\hfill \\ {}f(x)=\cos ax,\kern0.75em a\in \mathrm{\mathbb{C}}\wedge \mathrm{a}\ne 0\hfill & \int f(x) dx=\frac{1}{a}\sin ax+C\hfill \\ {}f(x)=\tan x,\hfill & \int f(x) dx=-\ln \left|\cos x\right|+\mathrm{C}\hfill \\ {}f(x)={a}^x,\kern0.5em a>0\wedge \mathrm{a}\ne 1,\hfill & \int f(x) dx=\frac{a^x}{\ln a}+C\hfill \\ {}f(x)=\frac{1}{\sqrt{x^2\pm 1}},\hfill & \int f(x) dx=\ln \left|x+\sqrt{x^2\pm 1}\right|+C\hfill \\ {}f(x)=\frac{1}{x^2+1},\hfill & \int f(x) dx=\arctan x+C\hfill \\ {}f(x)=\frac{1}{\sqrt{1-{x}^2}},\hfill & \int f(x) dx=\arcsin x+C\hfill \end{array}} $$

Basic rules of integration

• 1. $\frac{d}{dx}\int f(x) dx=f(x)$

• 2. $\int \frac{d}{dx}f(x) dx=f(x)+C$

• 3. ∫af(x)dx = a ∫ f(x)dx, if a is a constant

• 4. ∫[af(x) ± bg(x)]dx = a ∫ f(x)dx ± b ∫ g(x)dx, if a and b are constants (linearity).

Definite integral

$\underset{a}{\overset{b}{\int \limits }}f(x) dx$

where a and b are the limits of integration.

If F(x) =  ∫ f(x)dx then $\underset{a}{\overset{b}{\int \limits }}f(x) dx=F(b)-F(a)$ or $\underset{a}{\overset{b}{\int \limits }}f(x) dx={\left.F(x)\right|}_{x=a}-{\left.F(x)\right|}_{x=b}$

where F(x)| _x = a is F(x) for x=a.

Important rules for definite integrals

• 1. $\underset{a}{\overset{a}{\int \limits }}f(x) dx=0$

• 2. $\underset{a}{\overset{b}{\int \limits }}f(x) dx=-\underset{b}{\overset{a}{\int \limits }}f(x) dx$

• 3. If c ∈ (a, b) then $\underset{a}{\overset{b}{\int \limits }}f(x) dx=\underset{a}{\overset{c}{\int \limits }}f(x) dx+\underset{c}{\overset{b}{\int \limits }}f(x) dx$

Integration by parts

∫f(x)g?>′(x)dx = f(x)g(x) −  ∫ f?>′(x)g(x)dx

Reverse chain rule

∫h?>′(x)g?>′(h(x))dx = g(h(x)) + C

Expected value

a = 〈x〉 =  ∫ xP(x)dx

Convolution

$\left[f\ast g\right](t)=\underset{-\infty }{\overset{\infty }{\int \limits }}f\left(\tau \right)g\left(t-\tau \right) d\tau$

Cross-correlation

$f\ast g\left(\tau \right)=\underset{-\infty }{\overset{\infty }{\int \limits }}{f}^{\ast }(t)g\left(t+\tau \right) dt$

Answers to Exercises

• 7.1.

  • a. $\frac{1}{3}{e}^{3t}-\cos 2t+C$

  • b. $\frac{-3}{x}+C$

  • c. 4 ln |x| + C

  • d. $\frac{4}{3}{x}^{3/2}+C$

  • e. $-4{x}^{-\frac{1}{2}}+5x+C$

  • f. $\frac{3^x}{\ln 3}-\ln \left|\cos x\right|+C$

• 7.2.

  • a. $\frac{2}{5}$

  • b. $\frac{T}{\pi }$

  • c. $e\left(\frac{1}{3}{e}^2-e+1\right)-\frac{1}{3}$

• 7.3.

  • a. (x − 1)² e ^x − 2e ^x(x − 1) + 2e ^x + C = e ^x(x ² − 4x + 5) + C

  • b. $-\frac{\pi }{2}$

  • c. In this case, we suggest to use integration by parts twice. First, we write

  $f(x)={\left(\ln x\right)}^2,\kern0.75em {f}^{\prime }(x)=2\frac{1}{\mathrm{x}}\ln x$. That leaves us with g?>′(x) = 1, thus g(x) = x.

  So, applying integration by parts once, we find that:

  $\int {\left(\ln x\right)}^2 dx=x{\left(\ln x\right)}^2-\int \frac{2x}{x}\ln xdx=x{\left(\ln x\right)}^2-2\int \ln xdx$

  The remaining integral on the right-hand side, we can solve by again applying integration by parts. This time we choose f(x) = ln x, ${f}^{\hbox{'}}(x)=\frac{1}{x}$, and as above g?>′(x) = 1, thus g(x) = x.

  ∫ ln xdx = x ln x −  ∫ dx = x ln x − x + C

  So, we finally arrive at:

  ∫(lnx)² dx = x(lnx)² − 2x ln x + 2x + C

• 7.4.

  • a. $-\frac{1}{12}{e}^{-4{x}^3}+C$

  • b. −3 ln |2 + cos x| + C

  • c. $\frac{2}{7}{\left(\sqrt{x}+2\right)}^7+C$

  • d. 3 ln |ln|x||+C

  • e. $\frac{2}{3}\left(\sqrt{8}-1\right)$

• 7.5.

  • a. Use

    h(x) = x ⁴ ,  h'(x) = 4x ³,

    g'(h) = e ^h ,  g(h) = e ^h

  The result is $\frac{1}{4}{e}^{x^4} + C$

  • b. Use

    $h(x)=\frac{1}{4}{x}^4,\kern2em {h}^{\hbox{'}}(x)={x}^3,$

    ${g}^{\hbox{'}}(h)={\left(1+4h\right)}^3,\kern2em g(h)=\frac{1}{16}{\left(1+4h\right)}^4$

  The result is $\frac{1}{16}{\left(1+{x}^4\right)}^4 + C$

Glossary

Analytic

As in analytic expression, a mathematical expression that is written such that it can easily be calculated. Typically, it contains the basic arithmetic operations (addition, subtraction, multiplication, division) and operators such as exponents, logarithms and trigonometric functions.

Convolution

Convolution of a function or time series f(t), with another function g(t) yields the amount by which g(t) overlaps with f(t) when g(t) is shifted in time; convolution can be viewed as a modifying function or filter.

Cross-correlation

A measure of similarity of two functions as a function of the displacement of one with respect to the other.

Definite

As in definite integral; the integral of a function on a limited domain.

Double integral

Multiple integral with two variables of integration.

fMRI

Functional magnetic resonance imaging; a neuroimaging technique that employs magnetic fields and radiofrequency waves to take images of e.g. the functioning brain, employing that brain functioning is associated with changes in oxygenated blood flow.

Gaussian distribution

Also known as normal distribution. It is a symmetric, bell-shaped distribution that is very common and occurs when a stochastic variable is determined by many independent factors.

General linear model

Multiple linear regression; predicting a dependent variable from a set of independent variables according to a linear model.

Hemodynamic response function

A model function of the increase in blood flow to active brain neuronal tissue.

Indefinite

As in indefinite integral; the integral of a function without specification of a domain.

Integrand

Function that is integrated.

Inverse

As in ‘inverse operation’ or ‘inverse function’, meaning the operation or function that achieves the opposite effect of the original operation or function. For example, integration is the inverse operation of differentiation and lnx and e ^x are each other’s inverse functions.

Limit

Here: the boundaries of the domain for which the definite integral is determined.

Multiple integral

Definite integral over multiple variables.

Numerical integration

Estimating the value of a definite integral using computer algorithms.

Primitive

Anti-derivative.

Stochastic variable

A variable whose value depends on an outcome, for example the result of a coin toss, or of throwing a dice.

Triple integral

Multiple integral with three variables of integration.