Introduction to Integrals (With Examples)

If derivatives are about breaking things apart to see how fast they are changing, integrals are about putting those pieces back together to see how much has accumulated.

Imagine you are driving a car. Your speedometer tells you your exact speed at any given frozen moment—that is your derivative. But what if your speedometer is the only instrument working, and your odometer is broken? How do you figure out the total distance you have traveled? You have to multiply your speed by the time you spent driving. When speed is constantly changing, adding up all those tiny slivers of distance is called taking an integral.

Here is exactly what an integral is, the reverse-power rule used to calculate it, and several examples of how to find both indefinite and definite integrals.

What is an Integral?

In calculus, taking an integral (often called “integration” or finding the “antiderivative”) is literally the mathematical reverse of taking a derivative.

There are two main types of integrals you will encounter:

Indefinite Integrals: These give you a general formula (a new function) that represents the antiderivative. Because the derivative of any constant (like $+5$ or $-10$ ) is zero, taking the reverse step means we lose track of what that original constant might have been. To fix this, we always add a “$+ C$” at the end of an indefinite integral to represent that unknown constant.
Definite Integrals: These calculate a specific, numerical total between two exact points on a graph (from $a$ to $b$ ). Visually, this represents the exact area under the curve.

Before we calculate them by hand, you can visualize how mathematicians first solved this problem: by stuffing the area under a curve with tiny rectangles (called a Riemann Sum) and adding them up. The narrower the rectangles, the closer you get to the true integral:

The Reverse Power Rule

To take the derivative of $x^n$ , you multiply by $n$ and subtract 1 from the exponent. To take the integral, you do the exact opposite: add 1 to the exponent, then divide by the new exponent.
$\int x^n , dx = \frac{x^{n+1}}{n+1} + C$
(Note: The elongated “S” is the integral symbol, meaning “Sum”, and the “$dx$” simply tells you which variable you are integrating with respect to).

5 Examples of Finding Integrals

Let’s look at how to reverse-engineer derivatives using the reverse power rule and the Fundamental Theorem of Calculus.

Example 1: The Basic Indefinite Integral

Let’s find the integral of a basic polynomial piece:

$\int 3x^2 , dx$
Apply the reverse power rule. Add 1 to the exponent ( $2 + 1 = 3$ ). Then, divide the coefficient by that new exponent ( $3 / 3 = 1$ ):

$\frac{3x^3}{3} + C = x^3 + C$
You can always check your work by taking the derivative of your answer! The derivative of $x^3 + C$ is indeed $3x^2$ .

Example 2: Integrating Constants and Sums

Just like derivatives, you can integrate long polynomials piece by piece. What happens to a plain number?

$\int (4x^3 - 6x + 5) , dx$
Go term by term:

For $4x^3$ : Add 1 to the power (4), divide the coefficient by 4. Result: $x^4$ .
For $-6x^1$ : Add 1 to the power (2), divide $-6$ by 2. Result: $-3x^2$ .
For $5$ : A constant gains an $x$ (because the derivative of $5x$ is 5). Result: $5x$ .

Combine them and don’t forget the $+ C$ :

$x^4 - 3x^2 + 5x + C$

Example 3: The Definite Integral (Finding Exact Area)

Let’s find the exact area under the curve $y = 2x$ between $x = 1$ and $x = 4$ . This is written as a definite integral with bounds:

$\int_{1}^{4} 2x , dx$
Step 1: Find the antiderivative (you can drop the $+C$ for definite integrals because it cancels out anyway).

$\frac{2x^2}{2} = x^2$
Step 2: Apply the Fundamental Theorem of Calculus. Evaluate the antiderivative at the top bound (4), and subtract the antiderivative evaluated at the bottom bound (1):

$F(4) - F(1)$
$(4)^2 - (1)^2$
$16 - 1 = 15$
The exact area under the line between 1 and 4 is 15 square units.

Example 4: Trigonometric Integrals

Because integration is just derivatives in reverse, you can solve trig integrals by remembering your derivative rules.

$\int \cos(x) , dx$
Ask yourself: “What function has a derivative of $\cos(x)$ ?” The answer is $\sin(x)$ .

$\sin(x) + C$
Conversely, because the derivative of $\cos(x)$ is $-\sin(x)$ , integrating a positive sine requires a negative sign to balance it out:

$\int \sin(x) , dx = -\cos(x) + C$

Example 5: Fractional Exponents (Roots)

The reverse power rule works perfectly for roots and fractions. Let’s integrate a square root:

$\int \sqrt{x} , dx$
First, rewrite the root as a fractional exponent:

$\int x^{1/2} , dx$
Add 1 to the exponent ( $\frac{1}{2} + 1 = \frac{3}{2}$ ). Now, divide by the new exponent (which is the same as multiplying by its reciprocal, $\frac{2}{3}$ ):

$\frac{2}{3}x^{3/2} + C$

Why This Matters

While derivatives find the rate of change, integrals find the total accumulation.

If a factory’s computers know the rate at which water is leaking from a tank (gallons per minute), taking the integral tells them exactly how many total gallons were lost. If an engineer knows the varying force applied to move an object, the integral calculates the total work done. By allowing us to add up an infinite number of infinitely small pieces, the integral is the ultimate mathematical tool for measuring volume, area, mass, and total real-world impact.

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Introduction to Integrals (With Examples)