What does an indefinite integral represent?
Quick answer
An indefinite integral represents all functions whose derivative is the given function. If F′(x) = f(x), then the antiderivatives on the interval have the form F(x) + C: differentiating removes the constant. For example, ∫x² dx = x³/3 + C. The + C matters because every constant gives a different function with the same derivative. [S1]
The question, the details, the next step.
I know differentiation finds a slope or rate of change. When I see an indefinite integral, what is it asking me to find? Is it one particular function?
It asks you to find a function—or, more precisely, the family of functions—whose derivative is the expression inside the integral. Such a function is called an antiderivative. If F′(x) = f(x), then the antiderivatives on the interval can be written F(x) + C. The constant C is included because differentiating any constant gives zero, so adding a constant does not change the derivative. [S1]
The notation gives a general antiderivative, with the constant undetermined unless another condition supplies more information.
Can you work through a simple power example and show where the constant comes from?
Take ∫x² dx. For a power xⁿ with n ≠ −1, the power rule for antiderivatives says to increase the exponent by one and divide by the new exponent. Then include the integration constant: [S1]
∫x² dx = x³/3 + C.
Here, increasing 2 to 3 gives x³, and dividing by 3 gives x³/3. The constant records that there are many valid answers. For example, x³/3, x³/3 + 4, and x³/3 − 2 all have the same derivative. The power rule stated here excludes exponent −1. [S1]
How can I check that x³/3 + C is really an answer? Does the check tell me what C should be?
Differentiate your proposed antiderivative and see whether you get the original integrand:
- Differentiate x³/3 to get x².
- Differentiate C to get 0.
- Combine the results: d/dx(x³/3 + C) = x² + 0 = x².
That matches the integrand, so the antiderivative is verified. This check confirms the family of answers, but it does not select one value of C. A separate condition, such as a specified function value, would be needed to determine a particular constant. Without that condition, keep + C. [S1]
Could you show how I might write this as a complete solution, and what mistakes should I watch for?
A clear copyable solution is:
“Find ∫x² dx. By the power rule, increase the exponent by one and divide by 3: ∫x² dx = x³/3 + C. Check: d/dx(x³/3 + C) = x², which is the original integrand. Therefore, ∫x² dx = x³/3 + C.” [S1]
Two common omissions are leaving out + C and skipping the derivative check. Leaving out the constant gives only one member of the family, not the general antiderivative. In the check, remember that the constant disappears; it is not carried into the result as another nonzero term. The check establishes that the proposed derivative matches x², not that a particular constant has been chosen. [S1]
A practical checklist
Read the integrand as the derivative you need to reverse.
For a power xⁿ with n ≠ −1, add 1 to the exponent and divide by the new exponent. [S1]
Write + C to represent the family of antiderivatives. [S1]
Differentiate your answer, including the constant, and compare with the integrand. [S1]
Use an additional function-value condition if the task asks for one particular antiderivative. [S1]
A few more useful answers
Why can different functions all be antiderivatives of x²?
They can differ by a constant because differentiating a constant gives zero. Thus x³/3 + C has derivative x² for any constant C. [S1]
Why does the power rule exclude exponent −1?
The power rule stated here applies when n ≠ −1, so it cannot be used for that exponent. [S1]
How do I find C if an exercise gives a function value?
Substitute the stated input and output into the antiderivative, then solve the resulting equation for C. If no condition is given, leave the answer as a family with + C. [S1]
Sources
References used for this guide.
- [S1] OpenStax Calculus Volume 1, Section 4.10: Antiderivatives openstax.org · Checked