# Integral of a Polynomial

How to find the integral of a polynomial function: definition of an integral, formula, 2 examples, and their solutions.

## Integral

### Definition

Integral is an opposite operation of a derivative.

So, if the derivative of f(x) is f'(x),

the integral of f'(x) is f(x).

This is why

the integral is also called the antiderivative.

Then let's see how to write the integral of f(x).

The integral of f(x) is ∫ f(x) dx.

It is read as [integral f(x) d x].

Integral and Derivative are the opposte operations.

So the derivative of ∫ f(x) dx is f(x).

## Formula

### Formula

The derivative of [1/(n + 1)]x^{n + 1} + C is

[1/(n + 1)]⋅(n + 1)x^{n} = x^{n}.

Derivative of a Polynomial

And integral and derivative

are the opposite operations.

So the integral of x^{n} is

[1/(n + 1)]x^{n + 1} + C.

First write the reciprocal of n + 1: [1/(n + 1)].

Increase the exponent of the power: x^{n + 1}.

And write the constant term +C.

∫ f(x) dx doesn't have lower limit and upper limit.

So it's an indefinite integral.

For an indefinite integral,

there should be +C.

## Example 1

### Example

### Solution

The exponent of x^{3} is 3.

3 + 1 = 4

The reciprocal of 4 is 1/4.

So write 1/4.

3 + 1 = 4

So write x^{4}.

The given integral is an indefinite integral.

So write +C.

So [1/4]x^{4} + C is the answer.

## Example 2

### Example

### Solution

6 is the coefficient of 6x^{2}.

So write 6.

The exponent of x^{2} is 2.

2 + 1 = 3

The reciprocal of 3 is 1/3.

So write 1/3.

2 + 1 = 3

So write x^{3}.

-2 is the coeffiente of -2x.

So write -2.

x is x^{1}.

1 + 1 = 2

The reciprocal of 2 is 1/2.

So write 1/2.

1 + 1 = 2

So write x^{2}.

+5 is a constant term.

The integral of [constant] is [constant]x.

So the integral of +5 is +5x.

The given integral is an indefinite integral.

So write +C.

6⋅[1/3]x^{3} = 2x^{3}

-2⋅[1/2]x^{2} = -x^{2}

So 2x^{3} - x^{2} + 5x + C is the answer.