Integration

What is Integration in Maths?

  • 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, multiplication-division 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 anti-differentiation or integration.

  • Integration is a process of adding slices to find the whole.

  • It can be used to find areas, volumes, and central points.


Integration Definition Maths

\[\int_a^b {f\left( x \right)dx = } \]value of the anti-derivative at upper limit b – the value of the same anti-derivative at lower limit a.

 

Here What is Integration in Maths!

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 anti-derivative 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.

 

Here are a Few Points Listed Below in the Table That You Need to Keep in Mind!


  • Since the integral of a function isn’t definite, therefore it is generally referred to as indefinite integral.



  • We can never find the integral of a function at a point; we always find the integral of a given function in an interval.



  • Integral of a function is not unique; integrals of a function generally differ by numbers.

 

Integration is the Inverse of Differentiation!

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 \[{x^2}\].

 

Why do we Write Plus C in Integration Class 11?

  • 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.

 

 What is Integration in Maths?

  • 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 –

  1. Differential Calculus

  2. Integral Calculus

 

What do you mean by 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.

 

Different Types of Integrals in Mathematics-

Types of Integration-

Till now we have learned what Integration is. In mathematics, there are two types of Integrations or integrals -

  • Definite Integral

  • Indefinite Integral

 

What is a Definite Integral?

  • A definite integral is an integral that contains both the upper and the lower limits. 

  • Definite Integral is also known as Riemann Integral.

 

Representation of a Definite Integral-

\[\int_a^b {f\left( x \right)dx} \]

 

What is an Indefinite Integral?

  • An indefinite integral is an integral that does not contain the upper and the lower limits. 

  • Indefinite Integral is also known as Anti-Derivative or Primitive Integral.

  • Indefinite integral of a function f is generally a differentiable function F whose derivative is equal to the original function f.

 

Representation of a Definite Integral-

\[\int {f\left( x \right)} dx = f\left( x \right) + c\]

 

Some Elementary Standard Integrals in Integration Class 11 -

\[\int {{x^n}dx} \]

\[\frac{{{x^{n + 1}}}}{{n + 1}} + c{\text{ }}wheren \ne  - 1\]

\[\int {\sin x{\text{ }}dx} \]

\[ - {\text{ }}cos{\text{ }}x{\text{ }} + {\text{ }}C\]

\[\int {\cos x{\text{ }}dx} \]

\[sin{\text{ }}x{\text{ }} + {\text{ }}C\]

\[\int {{{\sec }^2}x{\text{ }}dx} \]

\[tan{\text{ }}x{\text{ }} + C\]

\[\int {\cos e{c^2}x{\text{ }}dx} \]

\[ - cot{\text{ }}x{\text{ }} + {\text{ }}C\]

\[\int {\sec x\tan x{\text{ }}dx} \]

\[sec{\text{ }}x{\text{ }} + {\text{ }}C\]

\[\int {\cos es{\text{ }}x\cot x{\text{ }}dx} \]

\[ - cosec{\text{ }}x{\text{ }} + C\]

 

Four Standard Theorems of Integration-

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

\[\int {f'\left( {g\left( x \right)} \right)g'\left( x \right)dx}  = f\left( {g\left( x \right)} \right) + c\]

 

Five Different Types of Integration Techniques-

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

 

Questions to be Solved

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

= t4/3 / (4/3) + C      

= (3/4) t4/3 + C