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Integrals - Review

This review page contains a summary of integration rules, that is, of rules for computing definite and indefinite integrals of a function.

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Indefinite integrals

If $fleft( x
ight) $ is a function of one variable, an indefinite integral of $fleft( x
ight) $ is a function $Fleft( x
ight) $ whose first derivative is equal to $fleft( x
ight) $:[eq1]An indefinite integral $Fleft( x
ight) $ is denoted by[eq2]Indefinite integrals are also called antiderivatives or primitives.

Example Let [eq3]The function[eq4]is an indefinite integral of $fleft( x
ight) $ because[eq5]Also the function[eq6]is an indefinite integral of $fleft( x
ight) $ because[eq7]

Note that if a function $Fleft( x
ight) $ is an indefinite integral of $fleft( x
ight) $ then also the function[eq8]is an indefinite integral of $fleft( x
ight) $ for any constant $cin U{211d} $ because[eq9]This is also the reason why the adjective indefinite is used: because indefinite integrals are defined only up to a constant.

The following subsections contain some rules for computing the indefinite integrals of functions that are frequently encountered in probability theory and statistics. In all these subsections, $c$ will denote a constant and the integration rules will be reported without a proof. Proofs are trivial and can be easily performed by the reader: it suffices to compute the first derivative of $Fleft( x
ight) $ and verify that it equals $fleft( x
ight) $.

Indefinite integral of a constant function

If $fleft( x
ight) $ is a constant function[eq10]where $ain U{211d} $, then an indefinite integral of $fleft( x
ight) $ is[eq11]

Indefinite integral of a power function

If $fleft( x
ight) $ is a power function[eq12]then an indefinite integral of $fleft( x
ight) $ is[eq13]when $n
eq -1$. When $n=-1$, that is, when[eq14]the integral is[eq15]

Indefinite integral of a logarithmic function

If $fleft( x
ight) $ is the natural logarithm of x, that is,[eq16]then its indefinite integral is[eq17]

If $fleft( x
ight) $ is the logarithm to base $b$ of x, that is,[eq18]then its indefinite integral is[eq19](remember that [eq20]).

Indefinite integral of an exponential function

If $fleft( x
ight) $ is the exponential function[eq21]then its indefinite integral is[eq22]

If the exponential function $fleft( x
ight) $ does not have the natural base $e$, but another positive base $b$, that is,[eq23]then its indefinite integral is[eq24](remember that [eq25]).

Indefinite integral of a linear combination of functions

If [eq26] and [eq27] are two functions and [eq28] are two constants, then[eq29]

In other words, the integral of a linear combination is equal to the linear combinations of the integrals. This property is called "linearity of the integral".

Two special cases of this rule are[eq30]

Indefinite integrals of trigonometric functions

The trigonometric functions have the following indefinite integrals:[eq31]

Definite integrals

Let $fleft( x
ight) $ be a function of one variable and $left[ a,b
ight] $ an interval of real numbers. The definite integral (or, simply, the integral) from a to $b$ of $fleft( x
ight) $ is the area of the region in the $xy$-plane bounded by the graph of [eq32], the x-axis and the vertical lines $x=a$ and $x=b$, where regions below the x-axis have negative sign and regions above the x-axis have positive sign.

The integral from a to $b$ of $fleft( x
ight) $ is denoted by[eq33]

$fleft( x
ight) $ is called the integrand function and a and $b$ are called upper and lower bound of integration.

The following subsections contain some properties of definite integrals, which are also often utilized to actually compute definite integrals.

Fundamental theorem of calculus

The fundamental theorem of calculus provides the link between definite and indefinite integrals. It has two parts.

On the one hand, if you define[eq34]then, the first derivative of $Fleft( x
ight) $ is equal to $fleft( x
ight) $, that is,[eq35]In other words, if you differentiate a definite integral with respect to its upper bound of integration, then you obtain the integrand function.

Example Define[eq36]Then,[eq37]

On the other hand, if $Fleft( x
ight) $ is an indefinite integral (an antiderivative) of $fleft( x
ight) $, then[eq38]

In other words, you can use the indefinite integral to compute the definite integral.

The following notation is often used:[eq39]where[eq40]

Sometimes the variable of integration x is explicitly specified and we write[eq41]

Example Consider the definite integral[eq42]The integrand function is[eq43]An indefinite integral of $fleft( x
ight) $ is[eq44]Therefore, the definite integral from 0 to 1 can be computed as follows.[eq45]

Definite integral of a linear combination of functions

Like indefinite integrals, also definite integrals are linear. If [eq46] and [eq27] are two functions and [eq48] are two constants, then[eq49]

with the two special cases[eq50]

Example For example,[eq51]

Change of variable

If $fleft( x
ight) $ and g(x) are two functions, then the integral[eq52]can be computed by a change of variable, with the variable[eq53]

The change of variable is performed in the following steps:

  1. Differentiate the change of variable formula[eq53]and obtain[eq55]

  2. Recompute the bounds of integration:[eq56]

  3. Substitute g(x) and [eq57] in the integral:[eq58]

Example The integral[eq59]can be computed performing the change of variable[eq60]By differentiating the change of variable formula, we obtain[eq61]The new bounds of integration are[eq62]Therefore the integral can be written as follows:[eq63]

Integration by parts

Let $fleft( x
ight) $ and g(x) be two functions and $Fleft( x
ight) $ and $Gleft( x
ight) $ their indefinite integrals. The following integration by parts formula holds:[eq64]

Example The integral[eq65]can be integrated by parts, by setting[eq66]An indefinite integral of $fleft( x
ight) $ is[eq67]and $Gleft( x
ight) $ is an indefinite integral of[eq68]or, said differently, [eq69] is the derivative of [eq70]. Therefore,[eq71]

Exchanging the bounds of integration

Given the integral [eq72]exchanging its bounds of integration is equivalent to changing its sign:[eq73]

Subdividing the integral

Given the two bounds of integration a and $b$, with $aleq b$, and a third point $m$ such that $aleq mleq b$, then[eq74]

Leibniz integral rule

Given a function of two variables [eq75] and the integral[eq76]where both the lower bound of integration a and the upper bound of integration $b$ may depend on $y$, under appropriate technical conditions (not discussed here) the first derivative of the function $Ileft( y
ight) $ with respect to $y$ can be computed as follows:[eq77]where [eq78] is the first partial derivative of [eq75] with respect to $y$.

Example The derivative of the integral[eq80]is[eq81]

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Compute the following integral:[eq82]

Hint: perform two integrations by parts.

Solution

By performing two integrations by parts, we obtain[eq83]Therefore,[eq84]which can be rearranged to yield[eq85]or[eq86]

Exercise 2

Use Leibniz integral rule to compute the derivative with respect to $y$ of the following integral:[eq87]

Solution

Leibniz integral rule is [eq88]We can apply it as follows:[eq89]

Exercise 3

Compute the following integral:[eq90]

Solution

This integral can be solved by using the change of variable technique:[eq91]

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