In mathematics, when we cannot perform general addition operations, we use integration to add values on a large scale.
There are various methods in mathematics to integrate functions.
Just like, addition subtraction, multiplicationdivision integration, and differentiation are also a pair of inverse functions.
According to integration definition maths, it is a process of finding functions whose derivative is given is named antidifferentiation or integration.
Integration is a process of adding slices to find the whole.
It can be used to find areas, volumes, and central points.
\[\int_a^b {f\left( x \right)dx = } \]value of the antiderivative at upper limit b – the value of the same antiderivative at lower limit a. 
If\[\frac{d}{{dx}}\left( {f\left( x \right)} \right) = f\left( x \right)\], then \[\int {f\left( x \right)} dx = f\left( x \right) + c\] The function F(x) is called antiderivative or integral or primitive of the given function f(x) and c is known as the constant of integration or the arbitrary constant. The function f(x) is called the integrand and f(x) dx is known as the element of integration. 



Finding an integral is the opposite of finding a derivative.
For example, let us find the integral of 2x.
\[\;2x = {x^2}\] 
Here, we know that the derivative of \[{x^2}\] is 2x and the integral is \[{x^2}\].
C is known as the Constant of Integration or Arbitrary Constant.
Let’s explain you with the help of an example,
The derivative of \[{x^2} + 44\;{\text{ }}is{\text{ }}\;2x\], and the derivative of \[{x^2} + 9\;\]is also 2x and it goes on. Since the derivative of a constant is always equal to zero.
So, just writing + C at the end we tend to wrap things up.
According to integration definition maths to find the whole, we generally add or sum up many parts to find the whole.
As we know, Integration is a reverse process of differentiation, which is a process where we reduce the functions into smaller parts.
To find the summation under a very large scale the process of integration is used.
We can use calculators for the calculation of small addition problems which is a very easy task to do. In problems where the limits reach infinity, we use integration methods, to sum up, many parts.
Differentiation and Integration both are important parts of calculus.
In Mathematics, we Know That There are two Major Types of Calculus –
Differential Calculus
Integral Calculus
Let us now talk about Integral Calculus,
Integral Calculus is the branch of calculus concerned with the application of integrals.
To see what differential calculus is, let us take an example of a slope of a line in a graph. In any graph, we can find the slope of a line, using the slope formula. What do we do when we need to find the slope of a curve?
The slope of the points in a curve varies and here differential calculus comes into the picture.
Till now we have learned what Integration is. In mathematics, there are two types of Integrations or integrals 
Definite Integral
Indefinite Integral
A definite integral is an integral that contains both the upper and the lower limits.
Definite Integral is also known as Riemann Integral.
\[\int_a^b {f\left( x \right)dx} \] 
An indefinite integral is an integral that does not contain the upper and the lower limits.
Indefinite Integral is also known as AntiDerivative or Primitive Integral.
Indefinite integral of a function f is generally a differentiable function F whose derivative is equal to the original function f.
\[\int {f\left( x \right)} dx = f\left( x \right) + c\] 
\[\int {{x^n}dx} \]  \[\frac{{{x^{n + 1}}}}{{n + 1}} + c{\text{ }}wheren \ne  1\] 
\[\int {\sin x{\text{ }}dx} \]  \[  {\text{ }}cos{\text{ }}x{\text{ }} + {\text{ }}C\] 
\[\int {\cos x{\text{ }}dx} \]  \[sin{\text{ }}x{\text{ }} + {\text{ }}C\] 
\[\int {{{\sec }^2}x{\text{ }}dx} \]  \[tan{\text{ }}x{\text{ }} + C\] 
\[\int {\cos e{c^2}x{\text{ }}dx} \]  \[  cot{\text{ }}x{\text{ }} + {\text{ }}C\] 
\[\int {\sec x\tan x{\text{ }}dx} \]  \[sec{\text{ }}x{\text{ }} + {\text{ }}C\] 
\[\int {\cos es{\text{ }}x\cot x{\text{ }}dx} \]  \[  cosec{\text{ }}x{\text{ }} + C\] 
Theorem 1  \[\frac{d}{{dx}}\left( {\int {f\left( x \right)} } \right)dx = f\left( x \right)\] 
Theorem 2  \[\int {\alpha {\text{ }}f\left( x \right)dx = \alpha \int {f\left( x \right)dx,f{\text{ }}or{\text{ }}all{\text{ }}\alpha \in R} } \] 
Theorem 3  \[\int {\left( {{f_1}\left( x \right) + {f_2}\left( x \right)  {f_3}\left( x \right)............} \right)} dx = \]. \[\int {\left( {{f_1}\left( x \right)dx + \int {{f_2}\left( x \right)dx  \int {{f_3}} \left( x \right)dx} } \right)} \] 
Theorem 4  \[\int {f'\left( {g\left( x \right)} \right)g'\left( x \right)dx} = f\left( {g\left( x \right)} \right) + c\] 
Here’s a list of Integration Methods –
Integration by Substitution
Integration by Parts
Integration by Partial Fraction
Integration of Some particular fraction
Integration Using Trigonometric Identities
Question 1) Evaluate the integral \[\int {{x^{49}}dx} \].
Solution) We know the formula for integration,
\[\int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}} + c{\text{ }}} where n \ne  1\] 
\[ = \frac{{{x^{49 + 1}}}}{{49 + 1}} + c\]
\[\frac{{{x^{50}}}}{{50}} + c\]
Question 2) Evaluate the integral \[\int {\sqrt[3]{t}dt} \] .
Solution) \[\int {\sqrt[3]{t}dt} \] can be written as \[\int {{t^{\frac{1}{3}}}} dt\]
Now, we know the formula for integration,
\[\int {{x^n}dx} = \frac{{{x^{n + 1}}}}{{n + 1}} + c,{\text{ }}Where n\ne  1\] 
= t^{(1/3+1)} / ((1/3) +1) + C
= t^{4/3} / (4/3) + C
= (3/4) t^{4/3 }+ C
Share your contact information
Vedantu academic counsellor will be calling you shortly for your Online Counselling session.