# Integrals - Review

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

## Indefinite integrals

If is a function of one variable, an indefinite integral of is a function whose first derivative is equal to :An indefinite integral is denoted byIndefinite integrals are also called antiderivatives or primitives.

Example Let The functionis an indefinite integral of becauseAlso the functionis an indefinite integral of because

Note that if a function is an indefinite integral of then also the functionis an indefinite integral of for any constant becauseThis 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, 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 and verify that it equals .

### Indefinite integral of a constant function

If is a constant functionwhere , then an indefinite integral of is

### Indefinite integral of a power function

If is a power functionthen an indefinite integral of iswhen . When , that is, whenthe integral is

### Indefinite integral of a logarithmic function

If is the natural logarithm of , that is,then its indefinite integral is

If is the logarithm to base of , that is,then its indefinite integral is(remember that ).

### Indefinite integral of an exponential function

If is the exponential functionthen its indefinite integral is

If the exponential function does not have the natural base , but another positive base , that is,then its indefinite integral is(remember that ).

### Indefinite integral of a linear combination of functions

If and are two functions and are two constants, then

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

### Indefinite integrals of trigonometric functions

The trigonometric functions have the following indefinite integrals:

## Definite integrals

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

The integral from to of is denoted by

is called the integrand function and and 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 definethen, the first derivative of is equal to , that is,In other words, if you differentiate a definite integral with respect to its upper bound of integration, then you obtain the integrand function.

Example DefineThen,

On the other hand, if is an indefinite integral (an antiderivative) of , then

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

The following notation is often used:where

Sometimes the variable of integration is explicitly specified and we write

Example Consider the definite integralThe integrand function isAn indefinite integral of isTherefore, the definite integral from to can be computed as follows.

### Definite integral of a linear combination of functions

Like indefinite integrals, also definite integrals are linear. If and are two functions and are two constants, then

with the two special cases

Example For example,

### Change of variable

If and are two functions, then the integralcan be computed by a change of variable, with the variable

The change of variable is performed in the following steps:

1. Differentiate the change of variable formulaand obtain

2. Recompute the bounds of integration:

3. Substitute and in the integral:

Example The integralcan be computed performing the change of variableBy differentiating the change of variable formula, we obtainThe new bounds of integration areTherefore the integral can be written as follows:

### Integration by parts

Let and be two functions and and their indefinite integrals. The following integration by parts formula holds:

Example The integralcan be integrated by parts, by settingAn indefinite integral of isand is an indefinite integral ofor, said differently, is the derivative of . Therefore,

### Exchanging the bounds of integration

Given the integral exchanging its bounds of integration is equivalent to changing its sign:

### Subdividing the integral

Given the two bounds of integration and , with , and a third point such that , then

### Leibniz integral rule

Given a function of two variables and the integralwhere both the lower bound of integration and the upper bound of integration may depend on , under appropriate technical conditions (not discussed here) the first derivative of the function with respect to can be computed as follows:where is the first partial derivative of with respect to .

Example The derivative of the integralis

## Solved exercises

Below you can find some exercises with explained solutions.

### Exercise 1

Compute the following integral:

Hint: perform two integrations by parts.

Solution

By performing two integrations by parts, we obtainTherefore,which can be rearranged to yieldor

### Exercise 2

Use Leibniz integral rule to compute the derivative with respect to of the following integral:

Solution

Leibniz integral rule is We can apply it as follows:

### Exercise 3

Compute the following integral:

Solution

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

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