Integral

Integral

There is a very thin line of distinction between integration, integral, and derivative. 

Let's try to understand it from the figure given here. 

Derivative is an inverse process of integration | Integral and Integration

In this topic, we will cover the basics of integration along with integral definition and types of integrals.

Lesson Plan

What Is the Meaning of Integral?

Given the derivative \(f’\) of the function \(f\), a question that arises is, "Can we determine the function\(f\)?"

Here, the function \(f\) is called antiderivative or integral of \(f’\).

The process of finding the antiderivative is called integration. On the other hand, the value of the function found by the process of integration is called an Integral

For example,

The derivative of \(f(x)=x^{3}\) is \(f’(x)=3x^{2}\);

And the antiderivative of \(g(x)=3x^{2}\) is \(f(x)=x^{3}\)

Here, the integral of \(g(x)=3x^{2}\) is \(f(x)=x^{3}\)


What Are the Different Types of Integral?

In general, there are two types of integrals.

Definite Integrals

These are the integrals that have a pre-existing value of limits; thus making the final value of integral definite. 

Indefinite Integrals

These are the integrals that do not have a pre-existing value of limits; thus making the final value of integral indefinite. 

Antiderivatives and indefinite integrals review (article) | Khan Academy

Here, \(c\) is the integration constant.


What Are the Integration Formulas?

We can remember the formulas of derivatives of some important functions.

Here are the corresponding integrals of these functions.

Derivatives Integrals
\(\dfrac{d}{dx}\left(\dfrac{x^{n+1}}{n+1}\right)=x^{n}\) \(\int x^{n} dx=\dfrac{x^{n+1}}{n+1}+C\), where \(n \neq -1\)
\(\dfrac{d}{dx}\left(x\right)=1\) \(\int  dx=x+C\)
\(\dfrac{d}{dx}\left(\sin{x}\right)=\cos{x}\) \(\int \cos{x}dx=\sin{x}+C\)
\(\dfrac{d}{dx}\left(-\cos{x}\right)=\sin{x}\) \(\int \sin{x}dx=-\cos{x}+C\)
\(\dfrac{d}{dx}\left(\tan{x}\right)=\sec^{2}{x}\) \(\int \sec^{2}{x}dx=\tan{x}+C\)
\(\dfrac{d}{dx}\left(-\cot{x}\right)=\csc^{2}{x}\) \(\int \csc^{2}{x}dx=-\cot{x}+C\)
\(\dfrac{d}{dx}\left(\sec{x}\right)=\sec{x}\tan{x}\) \(\int \sec{x} \tan{x}dx=\sec{x}+C\)
\(\dfrac{d}{dx}\left(-\csc{x}\right)=\csc{x}\cot{x}\) \(\int \csc{x} \cot{x}dx=-\csc{x}+C\)
\(\dfrac{d}{dx}\left(\sin^{-1}{x}\right)=\dfrac{1}{\sqrt{1-x^{2}}}\) \(\int\left(\dfrac{1}{\sqrt{1-x^{2}}}\right)dx=\sin^{-1}{x}+C\)
\(\dfrac{d}{dx}\left(\cos^{-1}{x}\right)=-\dfrac{1}{\sqrt{1-x^{2}}}\) \(\int\left(-\dfrac{1}{\sqrt{1-x^{2}}}\right)dx=\cos^{-1}{x}+C\)
\(\dfrac{d}{dx}\left(\tan^{-1}{x}\right)=\dfrac{1}{{1+x^{2}}}\) \(\int\left(\dfrac{1}{{1+x^{2}}}\right)dx=\tan^{-1}{x}+C\)
\(\dfrac{d}{dx}\left(\cot^{-1}{x}\right)=-\dfrac{1}{{1+x^{2}}}\) \(\int\left(-\dfrac{1}{{1+x^{2}}}\right)dx=\cot^{-1}{x}+C\)
\(\dfrac{d}{dx}\left(\sec^{-1}{x}\right)=\dfrac{1}{x\sqrt{x^{2}-1}}\) \(\int\left(\dfrac{1}{x\sqrt{x^{2}-1}}\right)dx=\sec^{-1}{x}+C\)
\(\dfrac{d}{dx}\left(\csc^{-1}{x}\right)=-\dfrac{1}{x\sqrt{x^{2}-1}}\) \(\int\left(-\dfrac{1}{x\sqrt{x^{2}-1}}\right)dx=\csc^{-1}{x}+C\)
\(\dfrac{d}{dx}\left(e^{x}\right)=e^{x}\) \(\int e^{x}dx=e^{x}+C\)
\(\dfrac{d}{dx}\left(\ln{|x|}\right)=\dfrac{1}{x}\) \(\int \dfrac{dx}{x}=\ln{|x|}+C\)
\(\dfrac{d}{dx}\left(\dfrac{a^{x}}{\ln{a}}\right)=a^{x}\) \(\int a^{x} dx=\dfrac{a^{x}}{\ln{a}}+C\)

Try out different integral formulas in the given integral calculator. You can enter different values of random functions, and you will get their integrals here: 

 
important notes to remember
Important Notes
  1. The value of the function found by the process of integration is called an integral.
  2. In general, there are two types of integrals:
       Definite Integrals (the value of the integral is definite)
       Indefinite Integrals (the value of the integral is indefinite)

Solved Examples

Here you can find more integral examples.

Example 1

 

 

Rachel wants to understand the difference between integral and integration.

Can you help her?

Solution

The process of finding the antiderivative is called integration.

On the other hand, the value of the function found by the process of integration is called an integral. 

Example 2

 

 

Hailey is working on her math assignment.

She is unable to solve one question.

The question says "suppose \(\dfrac{d}{dx}(f(x))=4x^3-\dfrac{3}{x^4}\) and \(f(2)=0\), find the function \(f(x)\)."

Can you help her solve this question?

Solution

Integrate on both sides of the equation \(\dfrac{d}{dx}(f(x))=4x^3-3x^{-4}\)

\[\begin{align}\int \dfrac{d}{dx}(f(x)) dx&=\int \left(4x^3-3x^{-4}\right)dx\\f(x)&=4\left(\dfrac{x^4}{4}\right)-\dfrac{3x^{-3}}{-3}+C\\f(x)&=x^4+\dfrac{1}{x^{3}}+C\end{align}\]

Use the condition \(f(2)=0\) to find the value of \(C\).

\[\begin{align}f(2)&=0\\2^4+\dfrac{1}{2^{3}}+C&=0\\16+\dfrac{1}{8}+C&=0\\C&=-\dfrac{129}{8}\end{align}\]

The function is \(f(x)=x^4+\dfrac{1}{x^{3}}-\dfrac{129}{8}\)
Example 3

 

 

Find the integral \(\int \dfrac{x^2+1}{x^2-5x+6}dx\)

Solution

Observe that the function \(\dfrac{x^2+1}{x^2-5x+6}\) is an improper rational function.

So, by long division, this can be written as \(\dfrac{x^2+1}{x^2-5x+6}=1+\dfrac{5x-5}{x^2-5x+6}dx\)

Long division method

By factorization, we have \(x^2-5x+6=(x-2)(x-3)\)

So, \(1+\dfrac{5x-5}{x^2-5x+6}=1+\dfrac{5x-5}{(x-2)(x-3)}\)

Let's proceed with partial fractions on \(\dfrac{5x-5}{(x-2)(x-3)}\)

\[\begin{align}\dfrac{5x-5}{(x-2)(x-3)}=\dfrac{A}{x-2}+\dfrac{B}{x-3}\end{align}\].

On comparing, we get, \(5x-5=A(x-3)+B(x-2)\)

From this, we have a set of two linear equations.

\[\begin{align}A+B&=5\\3A+2B&=5\end{align}\]

Solving linear equations

On solving these equations we get, \(A=-5\) and \(B=10\)

So \(1+\dfrac{5x-5}{x^2-5x+6}=1-\dfrac{5}{x-2}+\dfrac{10}{x-3}\;\;\;\;\;\;\; \cdots (1)\)

Integrate on both sides of equation (1)

\[\begin{align}&\int \left(1+\dfrac{5x-5}{x^2-5x+6}\right)dx\\&=\int \left(1-\dfrac{5}{x-2}+\dfrac{10}{x-3}\right)dx\\&=\int dx-5\int \left(\dfrac{dx}{x-2}\right)+10\int \left(\dfrac{dx}{x-3}\right)\\&=x-5\ln{|x-2|}+10\ln{|x-3|}+C\end{align}\]

\(\int \left(1+\dfrac{5x-5}{x^2-5x+6}\right)dx\)=\(x-5\ln{|x-2|}+10\ln{|x-3|}+C\)
 
 
Challenge your math skills
Challenging Questions

Find the indicated integral.
\[\begin{align}\int \dfrac{dx}{3x^2+13x-10}\end{align}\]

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 

 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of Integral. The math journey around Integral starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath. 

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs)

1. Can an integral have two answers?

Yes, an indefinite integral can have infinite answers depending upon the value of the constant term; while a definite integral will a constant value. 

2. What is a double integral used for?

A double integral is used in order to calculate the areas of regions, find the volumes given a surface or also the mean value of any given function in a plane region. 

3. What is an integral symbol called?

An integral symbol is \( \int \).

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