This review page contains a summary of integration rules, that is, of rules for computing definite and indefinite integrals of a function.
Table of contents
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 .
If is a constant functionwhere , then an indefinite integral of is
If is a power functionthen an indefinite integral of iswhen . When , that is, whenthe integral is
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 ).
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 ).
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
The trigonometric functions have the following indefinite 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.
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.
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.
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,
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:
Differentiate the change of variable formulaand obtain
Recompute the bounds of integration:
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:
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,
Given the integral exchanging its bounds of integration is equivalent to changing its sign:
Given the two bounds of integration and , with , and a third point such that , then
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
Below you can find some exercises with explained solutions.
Compute the following integral:
Hint: perform two integrations by parts.
By performing two integrations by parts, we obtainTherefore,which can be rearranged to yieldor
Use Leibniz integral rule to compute the derivative with respect to of the following integral:
Leibniz integral rule is We can apply it as follows:
Compute the following integral:
This integral can be solved by using the change of variable technique:
Please cite as:
Taboga, Marco (2021). "Integrals - Review", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/mathematical-tools/integrals-review.
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