 # 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.

### 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 $\int {\sqrt{t}dt}$ .

Solution) $\int {\sqrt{t}dt}$ 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