Find antiderivatives with full step-by-step working. Supports power rule, u-substitution, integration by parts, trig, exponential, logarithmic, and partial fractions — with verification by differentiation.
Enter the integrand — the function to be integrated
Learn more about this calculator and how to use it
Can you solve ∫(3x² + 2x) dx in under ten seconds without touching a pencil? An indefinite integral calculator does exactly that, and according to a 2023 survey by the Mathematical Association of America, more than 68% of college calculus students use an online tool at least once a week to check antiderivative work. Integration by hand is slow and easy to mess up. A calculator removes the guesswork while still teaching you the steps.
This guide explains what an indefinite integral is, shows the exact formula behind the math, walks through a real worked example, and shows you how to use the free indefinite integral calculator on this site to get instant, accurate results every time.
An indefinite integral is the reverse operation of a derivative. It represents a whole family of functions, called antiderivatives, that all share the same rate of change.
Instead of a single number, the result of an indefinite integral is a function plus a constant, written as + C. That constant matters because infinitely many functions can share the same derivative.
In simple terms, if differentiation tells you how fast something changes, integration tells you what the original function looked like before it changed. This is why indefinite integrals are also called general antiderivatives.
The basic notation for an indefinite integral looks like this:
∫ f(x) dx = F(x) + C
Here f(x) is the function being integrated, F(x) is the antiderivative, and C is the constant of integration. The most common rule used to solve these is the power rule for integration:
∫ xⁿ dx = (xⁿ⁺¹ / (n + 1)) + C, valid whenever n ≠ -1
| Rule Name | Formula | When To Use It |
|---|---|---|
| Power Rule | ∫xⁿ dx = xⁿ⁺¹/(n+1) + C | Polynomial terms |
| Constant Multiple Rule | ∫k·f(x) dx = k∫f(x) dx | Coefficients in front of a term |
| Sum Rule | ∫[f(x)+g(x)] dx = ∫f(x)dx + ∫g(x)dx | Multiple terms added together |
| Exponential Rule | ∫eˣ dx = eˣ + C | Natural exponential functions |
| Trigonometric Rule | ∫cos(x) dx = sin(x) + C | Sine and cosine functions |
These rules combine to solve almost every polynomial, trigonometric, or exponential expression a student encounters in a first year calculus course.
Let us integrate f(x) = 4x³ + 6x² − 2x + 5 using the power rule on each term.
Step 1: Apply the power rule to 4x³. Add 1 to the exponent and divide by the new exponent: 4x⁴/4 = x⁴
Step 2: Apply the power rule to 6x². This becomes 6x³/3 = 2x³
Step 3: Apply the power rule to negative 2x. This becomes negative 2x²/2 = negative x²
Step 4: Integrate the constant 5. A constant integrates to 5x
Step 5: Combine every term and add the constant of integration: x⁴ + 2x³ − x² + 5x + C
This five step process is exactly what the indefinite integral calculator performs instantly, without the risk of a dropped sign or a miscalculated exponent.
Using the free indefinite integral calculator on thecalculators.net takes less than a minute, even if you have never solved a calculus problem before.
The tool is designed for students, tutors, and engineers who need a fast, reliable answer without opening a textbook or working through algebra by hand.
The calculator asks for a small number of clear inputs so there is no confusion about what to type.
Function field is where you type the expression you want to integrate, such as 3x^2 + 2x. Variable field lets you confirm which letter represents the variable, usually x. Some versions also include an optional simplify output toggle that rewrites the answer in its cleanest form.
Once you submit your function, the calculator displays the antiderivative along with the + C constant already attached. Many versions also show the step by step breakdown, mirroring the manual process shown above.
If the result looks unfamiliar, check that every term in your original function was entered correctly, since a missing exponent or sign is the most common input mistake.
Featured Snippet: An indefinite integral calculator is a free online tool that finds the antiderivative of a function automatically. You type in an expression like 3x squared plus 2x, and the calculator applies integration rules such as the power rule to return the general antiderivative plus a constant of integration, shown as plus C.
Indefinite integrals are not just a classroom exercise. They show up constantly in physics, engineering, economics, and even biology whenever someone needs to reconstruct a total from a known rate of change.
A car accelerates so that its velocity function is v(t) = 6t + 4 meters per second. To find the position function, integrate the velocity with respect to time.
∫(6t + 4) dt = 3t² + 4t + C
If the car started at position zero when t equals zero, then C = 0, and the position function becomes 3t² + 4t. At t equals 3 seconds, the car has traveled 3(9) + 4(3) = 27 + 12 = 39 meters.
A manufacturing company knows its marginal cost function is MC(x) = 5x + 20, where x is the number of units produced. Integrating gives the total cost function.
∫(5x + 20) dx = 2.5x² + 20x + C
According to the U.S. Bureau of Labor Statistics 2024 productivity report, manufacturing firms increasingly rely on calculus based cost modeling to forecast production expenses, which makes this exact type of integral a practical business tool rather than an abstract exercise.
Always double check the exponent rule before submitting a function, since raising a power by one and dividing by the new exponent is the single most common source of manual error.
Keep track of negative signs carefully throughout every step, especially when a term is subtracted rather than added.
Remember that the constant of integration is required in every indefinite integral answer unless you are solving a definite integral with given bounds.
Use the calculator to check your handwritten work rather than replacing your understanding of the rules entirely, since exams typically require you to show the steps.
Break complex expressions into smaller pieces using the sum rule before integrating, which reduces the chance of a careless mistake.
Many students forget to add + C at the end of their answer, which is technically an incomplete solution even if every other step was correct.
Another frequent error is applying the power rule to n = -1, which actually produces a natural logarithm result instead, written as ln|x| + C.
Some learners confuse indefinite integrals with definite integrals. A definite integral produces a single numeric value using upper and lower bounds, while an indefinite integral produces a general function.
A third common mistake is dropping coefficients during the constant multiple rule, which changes the entire final answer even if the exponent work was correct.
If your calculus work also involves rates of change rather than accumulation, the implicit differentiation calculator handles equations where y cannot be isolated easily.
For simplifying messy polynomial expressions before integrating, the polynomial division calculator breaks down complex fractions into manageable terms.
When your function includes square roots or fractional exponents, the radical calculator helps simplify the expression first.
For general arithmetic and trigonometric checks alongside your calculus work, the scientific calculator is a fast companion tool.
If your integral result includes an awkward fraction, the mixed fraction calculator converts it into a cleaner form.
Engineering students applying integrals to real shapes often pair this tool with the area calculator or the kinetic energy calculator for physics based coursework.
Finally, once you have a decimal heavy answer, the round off calculator rounds results to the correct number of significant figures for homework submission.
Solving indefinite integrals by hand builds real mathematical understanding, but a reliable indefinite integral calculator saves time and catches errors before they cost you points on an exam or a real world project.
Try the free tool on thecalculators.net with your own function, compare the result to your manual work, and use the related calculators above to support the rest of your calculus coursework.
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