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)]xn + 1 + C is
[1/(n + 1)]⋅(n + 1)xn = xn.

Derivative of a Polynomial

And integral and derivative
are the opposite operations.

So the integral of xn is
[1/(n + 1)]xn + 1 + C.

First write the reciprocal of n + 1: [1/(n + 1)].
Increase the exponent of the power: xn + 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 x3 is 3.
3 + 1 = 4
The reciprocal of 4 is 1/4.
So write 1/4.

3 + 1 = 4
So write x4.

The given integral is an indefinite integral.
So write +C.

So [1/4]x4 + C is the answer.

Example 2

Example

Solution

6 is the coefficient of 6x2.
So write 6.

The exponent of x2 is 2.
2 + 1 = 3
The reciprocal of 3 is 1/3.
So write 1/3.

2 + 1 = 3
So write x3.

-2 is the coeffiente of -2x.
So write -2.

x is x1.
1 + 1 = 2
The reciprocal of 2 is 1/2.
So write 1/2.

1 + 1 = 2
So write x2.

+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]x3 = 2x3
-2⋅[1/2]x2 = -x2

So 2x3 - x2 + 5x + C is the answer.