# Q Integral Of Constant With Respect To X Is, Integration And Differential Equations

**Q Integral Of Constant With Respect To X Is, Integration And Differential Equations**in

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## Basic Integration Principles

Integration is the process of finding the region bounded by a function; this process makes use of several important properties.

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### Key Takeaways

Key PointsThe term integral may also refer to the notion of the anti- derivative, a function **integration**: the operation of finding the region in the x-y plane bound by the function

Integration is an important concept in mathematics and—together with its inverse, differentiation—is one of the two main operations in calculus. Given a function

**Definite Integral**: A definite integral of a function can be represented as the signed area of the region bounded by its graph.

More rigorously, once an anti-derivative

If

Integration proceeds by adding up an infinite number of infinitely small areas. This sum can be computed by using the anti-derivative.

### Properties

### Linearity

The integral of a linear combination is the linear combination of the integrals.

### Inequalities

If

### Additivity

If

### Reversing Limits of Integration

If

### Integration by Substitution

By reversing the chain rule, we obtain the technique called integration by substitution. Given two functions

or written in terms of the “dummy variable”

If we are going to use integration by substitution to calculate a definite integral, we must change the upper and lower bounds of integration accordingly.

## Integration By Parts

Integration by parts is a way of integrating complex functions by breaking them down into separate parts and integrating them individually.

### Key Takeaways

Key PointsIntegration by parts is a theorem that relates the integral of a product of functions to the integral of their derivative and anti-derivative.The theorem is expressed as **integral**: also sometimes called antiderivative; the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed**derivative**: a measure of how a function changes as its input changes

### Introduction

In calculus, integration by parts is a theorem that relates the integral of a product of functions to the integral of their derivative and anti-derivative. It is frequently used to find the anti-derivative of a product of functions into an ideally simpler anti-derivative. The rule can be derived in one line by simply integrating the product rule of differentiation.

### Theorem of integration by parts

Let’s take the functions

or, more compactly,

### Proof

Suppose

ight) = u(x) frac{d}{dx}left(v(x)

ight) + frac{d}{dx}left(u(x)

ight) v(x)}

Integrating both sides with respect to

ight),dx = int_a^b u”(x)v(x),dx + int_a^b u(x)v”(x),dx}

then applying the fundamental theorem of calculus,

ight),dx = left___a^b}__

gives the formula for “integration by parts”:

___a^b = int_a^b u”(x)v(x),dx + int_a^b u(x)v”(x),dx}__

### Visulization

Let’s define a parametric curve by

**Integration By Parts**: Integration by parts may be thought of as deriving the area of the blue region from the total area and that of the red region. The area of the blue region is

### Example

In order to calculate

and

then:

## Trigonometric Integrals

The trigonometric integrals are a specific set of functions used to simplify complex mathematical expressions in order to evaluate them.

### Key Takeaways

Key PointsSome of the expressions for the trigonometric integrals are found using properties of trigonometric functions.Some of the expressions were derived using techniques like integration by parts.There is no guarantee that a trigonometric integral has an analytic expression.Key Terms**trigonometric**: relating to the functions used in trigonometry: **integral**: also sometimes called antiderivative; the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed

### Trigonometric Integrals

The trigonometric integrals are a family of integrals which involve trigonometric functions (

Generally, if the function,

In all formulas, the constant

### Integrands Involving Only Sine:

ight) sin 2ax – frac{x}{4a^2} cos 2ax +C!}

### Integrands Involving Only Cosine:

ight) sin 2ax + frac{x}{4a^2} cos 2ax +C

### Integrands Involving Only Tangent:

ight|+C\ & = frac{1}{a}ln left|sec ax

ight|+Cend{align}

where

eq 1

ight|)+C }

where

eq 0

### Integrands Involving Only Secant:

ight|}+C}

### Integrands involving only cosecant:

ight|}+C}

### Complicated Trigonometric Integrals

We now look at integrals involving the product of a power of

If

## Trigonometric Substitution

Trigonometric functions can be substituted for other expressions to change the form of integrands and simplify the integration.

### Key Takeaways

Key PointsIf the integrand contains **trigonometric**: relating to the functions used in trigonometry:

Trigonometric functions can be substituted for other expressions to change the form of integrands. One may use the trigonometric identities to simplify certain integrals containing radical expressions (or expressions containing

### Substitution Rule #1

If the integral contains

### Substitution Rule #2

If the integrand contains

### Substitution Rule #3

If the integrand contains

Note that, for a definite integral, one must figure out how the bounds of integration change due to the substitution.

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### Examples

In order to better understand these substitutions, let’s go over the derivation of some of them.

### Example 1: Integrals where the integrand contains a^2 − x^2 (where a is positive)

In the integral

we may use:

ight)}

With the substitution, we get:

ight)}+C end{align}

### Example 2: Integrals where the integrand contains a^2 − x^2 (where a is not zero)

In the integral

we may use:

ight)}

With the substitution, we get:

ight)+Cend{align}

## The Method of Partial Fractions

Partial fraction expansions provide an approach to integrating a general rational function.

### Key Takeaways

Key PointsAny rational function of a real variable can be written as the sum of a polynomial and a finite number of rational fractions whose denominator is the power of an irreducible polynomial and whose numerator has a degree lower than the degree of this irreducible polynomial.The substitution

ight|+C**irreducible**: unable to be factorized into polynomials of lower degree, as

Partial fraction expansions provide an approach to integrating a general rational function. Any rational function of a real variable can be written as the sum of a polynomial and a finite number of rational fractions whose denominator is the power of an irreducible polynomial and whose numerator has a degree lower than the degree of this irreducible polynomial. Here are some common examples.

### A 1st-Degree Polynomial in the Denominator

The substitution

ight|+C \ &= {1 over a} lnleft|ax+b

ight|+Cend{align}

### A Repeated 1st-Degree Polynomial in the Denominator

The same substitution reduces such integrals as

### An Irreducible 2nd-Degree Polynomial in the Denominator

Next we consider integrals such as

The quickest way to see that the denominator,

and observe that this sum of two squares can never be

we would need to find

The substitution handles the first summand, thus:

ight|+C \ &= frac{1}{2}ln(x^2-8x+25)+Cend{align}

Note that the reason we can discard the absolute value sign is that, as we observed earlier,

Next we must treat the integral

With a little more algebra,

ight)^2+1},dx \ &= {10 over 3} arctanleft(frac{x-4}{3}

ight) + C end{align}

Putting it all together:

ight) + C}

## Integration Using Tables and Computers

Tables of known integrals or computer programs are commonly used for integration.

### Learning Objectives

Recognize which integrals should be solved using tables or computers due to their complexity

### Key Takeaways

Key PointsWhile differentiation has easy rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not.In books with integral tables, a compilation of a list of integrals and techniques of integral calculus can be found.There are several commercial softwares, such as Mathematica or Matlab, that can perform symbolic integration.Key Terms**integral**: also sometimes called antiderivative; the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed

Integration is the basic operation in integral calculus. While differentiation has easy rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. We also may have to resort to computers to perform an integral.

### Integration Using Tables

A compilation of a list of integrals and techniques of integral calculus was published by the German mathematician Meyer Hirsch as early as in 1810. More extensive tables were compiled in 1858 by the Dutch mathematician David de Bierens de Haan. A new edition was published in 1862. These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century. They were then replaced by the much more extensive tables of Gradshteyn and Ryzhik. Here are a few examples of integrals in these tables for logarithmic functions:

ight| + ln x + sum^infty_{k=2}frac{(ln x)^k}{kcdot k! }}

eq 1mbox{)}}

You can certainly see that these integrals are hard to do simply “by hand.”

### Integration Using Computers

Computers may be used for integration in two primary ways. First, numerical methods using computers can be helpful in evaluating a definite integral. There are many methods and algorithms. We will briefly learn about numerical integration in another atom. Second, there are several commercial softwares, such as Mathematica or Matlab, that can perform symbolic integration.

### Key Takeaways

Key PointsThe trapezoidal rule works by approximating the region under the graph of the function **trapezoid**: a (convex) quadrilateral with two (non-adjacent) parallel sides

Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite integral. Popular methods use one of the Newton–Cotes formulas (such as midpoint rule or Simpson’s rule) or Gaussian quadrature. These methods rely on a “divide and conquer” strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use other methods such as the Monte Carlo method. Here, we will study a very simple approximation technique, called a trapezoidal rule.

### Trapezoidal rule

The trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral

The trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods.

### Numerical Implementation of the Trapezoidal Rule

For a domain discretized into

ight) {} \ &= frac{b-a}{2N}(f(x_1) + 2f(x_2) + cdots + 2f(x_N) + f(x_{N+1}))end{align}

Although the method can adopt a nonuniform grid as well, this example used a uniform grid for the the approximation.

## Improper Integrals

An Improper integral is the limit of a definite integral as an endpoint of the integral interval approaches either a real number or

### Learning Objectives

Evaluate improper integrals with infinite limits of integration and infinite discontinuity

### Key Takeaways

Key PointsAn improper integral may be a limit of the form **integrand**: the function that is to be integrated**definite integral**: the integral of a function between an upper and lower limit

An improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or

Specifically, an improper integral is a limit of one of two forms.

First, an improper integral could be a limit of the form:

Integrals are also improper if the integrand is undefined at an interior point of the domain of integration, or at multiple such points. It is often necessary to use improper integrals in order to compute a value for integrals which may not exist in the conventional sense (as a Riemann integral, for instance) because of a singularity in the function, or an infinite endpoint of the domain of integration.

### Example 1

The original definition of the Riemann integral does not apply to a function such as

ight) \ &= 1end{align}

### Example 2

The narrow definition of the Riemann integral also does not cover the function

## Numerical Integration

Numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral.

### Key Takeaways

Key PointsThe basic problem considered by numerical integration is to compute an approximate solution to a definite integral: **trapezoidal**: in the shape of a trapezoid, or having some faces which have one pair of parallel sides**antiderivative**: an indefinite integral

Numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and, by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals. Numerical integration over more than one dimension is sometimes described as cubature, although the meaning of quadrature is understood for higher dimensional integration as well.

The basic problem considered by numerical integration is to compute an approximate solution to a definite integral:

If

### Reasons for numerical integration

There are several reasons for carrying out numerical integration. The integrand

### Methods for One-Dimensional Integrals

A large class of quadrature rules can be derived by constructing interpolating functions which are easy to integrate. Typically these interpolating functions are polynomials. The simplest method of this type is to let the interpolating function be a constant function (a polynomial of degree zero) which passes through the point

ight)

ight)

ight)}

The interpolating function may be an affine function (a polynomial of degree 1) which passes through the points

For either one of these rules, we can make a more accurate approximation by breaking up the interval

ight)

ight) + {f(b) over 2}

ight)}

where the subintervals have the form

**Integral**