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Learn Integrals

Add up infinitely many tiny pieces to find areas and totals.

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An integral adds up a quantity that changes continuously. The definite integral of a function from a to b is the signed area under its curve, found as the limit of Riemann sums of thin rectangles. The indefinite integral, or antiderivative, runs differentiation in reverse. The Fundamental Theorem of Calculus links the two: to compute an area, find any antiderivative and subtract its values at the endpoints. Integration is harder than differentiation because there is no single rule for every case, so you learn a set of techniques and when to use each. Learnr builds a focused integrals course and teaches each technique in short interactive lessons with worked steps.

How to learn Integrals, step by step

Estimate areas with rectangles first. Approximate the area under x squared from 0 to 2 with four rectangles, then eight, and watch the sums close in on 8/3. This makes the definite integral a concrete limit rather than a symbol.
Learn antiderivatives by reversing derivatives. Since the derivative of x cubed is 3x squared, an antiderivative of x squared is x cubed over 3. Always add the constant C on indefinite integrals and check by differentiating your answer.
Use the Fundamental Theorem on every definite integral. Find an antiderivative F, then compute F(b) minus F(a). For x squared from 0 to 2, that is 8/3 minus 0. Practice with negative regions to see why the area is signed.
Master u-substitution. Look for an inner function whose derivative also appears. In the integral of 2x cos(x squared), set u = x squared, so du = 2x dx and the answer is sin(x squared) plus C. Remember to change limits on definite integrals.
Add integration by parts and partial fractions. Integration by parts reverses the product rule: for x times e to the x, let u = x and get x e to the x minus e to the x plus C. Partial fractions split rational functions into simpler pieces.
Mix techniques in short daily sets. The hardest part is deciding which method fits. Daily mixed practice in a Learnr course, kept steady with a streak, trains that decision faster than doing twenty of the same type in one sitting.

What you'll learn

Approximate areas with left, right and midpoint Riemann sums
Find antiderivatives of polynomial, trig and exponential functions
Evaluate definite integrals with the Fundamental Theorem of Calculus
Use u-substitution and integration by parts
Compute areas between curves and volumes of revolution

What's inside your Integrals course

A sample path Learnr builds for Integrals. Your real course adapts to your goal and level.

1. Area, Riemann sums and the definite integral
2. Antiderivatives and basic integral rules
3. The Fundamental Theorem of Calculus
4. U-substitution
5. Integration by parts and partial fractions
6. Applications: area, volume and accumulated change

How long does it take to learn Integrals?

If you already know derivatives, basic integration and the Fundamental Theorem take about 3 to 5 weeks at 3 to 5 hours a week. Adding techniques like parts, partial fractions and trig substitution, plus applications, takes roughly 6 to 10 more weeks.

What you need before you start

A solid grasp of derivatives, including the chain rule, plus algebra and trigonometric identities. Integration is differentiation in reverse, so derivative fluency matters most.

Common mistakes to avoid

Dropping the + C on indefinite integrals. Every antiderivative is a whole family of functions that differ by a constant.
Forgetting to convert the bounds after a u-substitution on a definite integral, or mixing x-bounds with an answer written in u.
Assuming integrals of products or quotients split into separate integrals. The integral of fg is not the integral of f times the integral of g.

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Integrals questions, answered

Why is integration harder than differentiation?

Differentiation has rules that always work on combinations of elementary functions. Integration has no such complete rulebook, and some simple-looking integrals, like e to the minus x squared, have no elementary antiderivative at all.

What is the difference between a definite and an indefinite integral?

A definite integral has bounds and gives a number, such as an area. An indefinite integral has no bounds and gives a family of functions, the antiderivatives, written with + C.