The best examples of definite and indefinite integrals explained step by step

Starting with real examples of definite and indefinite integrals explained

Let’s skip the abstract talk and jump straight into examples. Once you see how these work in practice, the vocabulary will feel much less intimidating.

Picture this: a car’s velocity (in feet per second) is given by the function
\(v(t) = 4t\), where \(t\) is time in seconds.

If you’re given the definite integral
\[\int_0^5 4t\,dt,\]
that means: “Find the exact change in position of the car from \(t = 0\) to \(t = 5\) seconds.”

If you’re given the indefinite integral
\[\int 4t\,dt,\]
that means: “Find all possible position functions whose derivative is \(4t\).”

Same formula under the integral sign, two different jobs. That contrast is at the heart of any good collection of examples of definite and indefinite integrals explained carefully.

Classic algebraic example of a definite integral (area under a curve)

Take the function \(f(x) = x^2\). A very standard question is:
\[\int_0^2 x^2\,dx.\]

This is a definite integral. The numbers 0 and 2 are limits of integration. Here’s how to handle it step by step:

First, find an antiderivative of \(x^2\):
\[\int x^2\,dx = \frac{x^3}{3} + C.\]

Now use it to evaluate the definite integral:
\[\int_0^2 x^2\,dx = \left.\frac{x^3}{3}\right|_0^2 = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3}.\]

Geometrically, \(\frac{8}{3}\) is the area under the curve \(y = x^2\) from \(x = 0\) to \(x = 2\). This is one of the best examples of how a definite integral turns a function into a single numerical answer.

Now contrast that with the related indefinite version.

Indefinite integral of the same function: family of antiderivatives

For the same \(f(x) = x^2\), the indefinite integral
\[\int x^2\,dx\]
asks you for all functions whose derivative is \(x^2\).

We already saw the antiderivative:
\[\int x^2\,dx = \frac{x^3}{3} + C.\]

Here \(C\) is a constant, and every different value of \(C\) gives a different function. They all have the same derivative, \(x^2\). That’s why an indefinite integral never gives just one function — it gives a whole family.

This pair — \(\int_0^2 x^2\,dx\) and \(\int x^2\,dx\) — is a simple but powerful example of definite and indefinite integrals explained side by side.

Examples of definite and indefinite integrals explained with motion

Let’s go back to that car with velocity \(v(t) = 4t\) feet per second.

Indefinite integral (position function)

To find a position function \(s(t)\) whose derivative is \(v(t) = 4t\):
\[s(t) = \int 4t\,dt = 2t^2 + C.\]

If you’re told the car starts at position \(s(0) = 10\) feet, you can solve for \(C\):
\[10 = 2(0)^2 + C \Rightarrow C = 10.\]
So the specific position function is
\[s(t) = 2t^2 + 10.\]

Definite integral (distance traveled)

Now suppose you want the distance traveled from \(t = 0\) to \(t = 5\) seconds. That’s a definite integral:
\[\int_0^5 4t\,dt = \left.2t^2\right|_0^5 = 2(25) - 0 = 50 \text{ feet}.\]

Same integrand, but the indefinite integral gave you a formula for position, while the definite integral gave you a number — the actual distance traveled over a time interval.

This kind of motion problem is one of the best examples of definite and indefinite integrals explained in physics and engineering courses.

Real examples of definite and indefinite integrals in everyday-style problems

To make the idea stick, let’s walk through several different types of examples. These examples of definite and indefinite integrals explained in context will help you recognize patterns when you see them on homework or exams.

Example 1: Indefinite integral of a polynomial

Find the indefinite integral:
\[\int (3x^2 - 4x + 7)\,dx.\]

Integrate term by term:
\[\int 3x^2\,dx = x^3,\]
\[\int (-4x)\,dx = -2x^2,\]
\[\int 7\,dx = 7x.\]

So:
\[\int (3x^2 - 4x + 7)\,dx = x^3 - 2x^2 + 7x + C.\]

This is a pure indefinite integral example: no limits, answer is a family of functions.

Example 2: Definite integral of a rate (water flowing into a tank)

Say water flows into a tank at a rate of
\[r(t) = 5 + 2t \quad \text{liters per minute},\]
for \(0 \le t \le 10\) minutes.

The total amount of water added in that time is
\[\int_0^{10} (5 + 2t)\,dt.\]

Compute an antiderivative:
\[\int (5 + 2t)\,dt = 5t + t^2 + C.\]

Now evaluate the definite integral:
\[\int_0^{10} (5 + 2t)\,dt = \left.(5t + t^2)\right|_0^{10} = (50 + 100) - 0 = 150.\]

So 150 liters of water are added during those 10 minutes. This is a nice real-world example of a definite integral turning a rate into a total amount.

Example 3: Indefinite integral with a trigonometric function

Find
\[\int \cos x\,dx.\]

We know that the derivative of \(\sin x\) is \(\cos x\), so
\[\int \cos x\,dx = \sin x + C.\]

If you’re later asked for
\[\int_0^{\pi} \cos x\,dx,\] you would use the same antiderivative but evaluate from 0 to \(\pi\): \[\int_0^{\pi} \cos x\,dx = \left.\sin x\right|_0^{\pi} = \sin(\pi) - \sin(0) = 0 - 0 = 0.\]

This pair is another clear example of definite and indefinite integrals explained using trig functions.

Example 4: Definite integral as average value of a function

Suppose the temperature (in degrees Fahrenheit) of a room over 8 hours is modeled by
\[T(t) = 72 + 4\sin\left(\frac{\pi t}{8}\right), \quad 0 \le t \le 8,\]
where \(t\) is time in hours.

The average temperature over those 8 hours is
\[T_{\text{avg}} = \frac{1}{8} \int_0^8 T(t)\,dt.\]

Even if you don’t fully work out the integral, this is a real example of a definite integral being used to compute an average over time — something that shows up in climate studies, building energy modeling, and more. Universities like MIT and Harvard use similar examples in their open calculus materials, which you can browse through on sites like MIT OpenCourseWare and Harvard Online Learning.

Examples of definite and indefinite integrals explained with substitution

Sometimes you need a small trick called substitution to make an integral manageable.

Example 5: Indefinite integral using substitution

Find
\[\int 2x\cos(x^2)\,dx.\]

Notice that the derivative of \(x^2\) is \(2x\), which is sitting right there. Let
\[u = x^2 \Rightarrow du = 2x\,dx.\]

Then the integral becomes
\[\int \cos(u)\,du = \sin(u) + C = \sin(x^2) + C.\]

This is a classic example of an indefinite integral explained with substitution.

Example 6: Definite integral using substitution

Now consider
\[\int_0^1 2x\cos(x^2)\,dx.\]

Use the same substitution \(u = x^2\), \(du = 2x\,dx\). When \(x = 0\), \(u = 0\). When \(x = 1\), \(u = 1\). The integral becomes
\[\int_0^1 \cos(u)\,du = \left.\sin(u)\right|_0^1 = \sin(1) - 0 = \sin(1).\]

Here you see another pair of examples of definite and indefinite integrals explained with exactly the same technique, just with or without limits.

A quick look at modern data and science uses (2024–2025 context)

Integrals are not just textbook exercises. In 2024–2025, they’re baked into the software and models behind everything from climate forecasts to medical imaging.

For example, when researchers at institutions like the National Institutes of Health (NIH) or Mayo Clinic analyze how a drug concentration changes in the bloodstream over time, they often use definite integrals to compute total exposure to the drug. The model might give a concentration function \(C(t)\), and a definite integral like
\[\int_0^{24} C(t)\,dt\]
represents the area under the concentration–time curve over 24 hours.

Behind the scenes, numerical methods approximate these integrals from real patient data. Even if you never see the formulas, the same ideas you practice in class — the kinds of examples of definite and indefinite integrals explained above — are at work in real-world research.

In data science and machine learning, optimization algorithms often use integrals to measure loss functions over continuous distributions. Universities and research labs, such as those represented on arXiv.org, routinely publish work where integrals define probabilities, expectations, or costs.

How to tell quickly: definite vs. indefinite

As you work through more examples of definite and indefinite integrals explained in your textbook or online, train your eye to look for two things.

If you see no limits:
\[\int f(x)\,dx,\]
you’re doing an indefinite integral. Your answer should be a function plus a constant \(+C\).

If you see limits: \[\int_a^b f(x)\,dx,\] you’re doing a *definite integral. Your answer should be a *number (possibly involving symbols like \(\pi\) or \(e\)), not a function of \(x\).

Many of the best examples of definite and indefinite integrals explained in class combine both: first find the indefinite integral (the antiderivative), then use it to evaluate a definite integral.

FAQ: common questions about examples of definite and indefinite integrals

Why do indefinite integrals always have “+ C” at the end?

Because an indefinite integral represents all antiderivatives of a function. If \(F(x)\) is one antiderivative of \(f(x)\), then \(F(x) + C\) is also an antiderivative for any constant \(C\). Without the \(+C\), you’re leaving out infinitely many valid answers.

Can you give another quick example of a definite integral that represents area?

Yes. Consider \(f(x) = 3x\) on the interval \([0, 4]\). The definite integral
\[\int_0^4 3x\,dx = \left.\frac{3x^2}{2}\right|_0^4 = \frac{3\cdot 16}{2} = 24\]
represents the area under the line \(y = 3x\) from \(x = 0\) to \(x = 4\). This is a simple example of a definite integral where the graph forms a triangle-like region whose area you could also compute geometrically.

How are these examples used in real college courses?

If you look at syllabi from U.S. universities like MIT or Harvard, you’ll see that early calculus weeks are packed with examples of definite and indefinite integrals explained using polynomials, exponentials, and trigonometric functions. They then move into applications: areas, volumes, work, probability, and averages. The goal is exactly what we’ve done here — start with concrete problems, then connect them to bigger ideas in science and engineering.

Do I always need to find the indefinite integral first to compute a definite integral?

For most standard problems in a first calculus course, yes: you find an antiderivative (an indefinite integral) and then plug in the limits. However, there are special techniques like numerical integration (trapezoid rule, Simpson’s rule) and some advanced tricks where you approximate the definite integral directly without ever writing down a closed-form antiderivative.

Where can I practice more examples of definite and indefinite integrals explained step by step?

Many universities and educational organizations offer free problem sets. You can explore calculus materials from MIT OpenCourseWare, open resources linked through Harvard.edu, or general math help sites that walk through example of integrals with full solutions. Look specifically for sections labeled “integrals,” “applications of integration,” or “antiderivatives” to find more examples like the ones in this guide.

If you keep revisiting these examples of definite and indefinite integrals explained in different contexts — motion, area, rates, averages — you’ll start to see the same patterns over and over. That’s when integrals stop feeling mysterious and start feeling like just another tool in your math toolbox.