Calculating Indefinite Integrals: Rules and Examples

Calculating an indefinite integral means finding a family of functions whose derivative equals the given integrand. Enter a function of x, and this calculator returns an antiderivative using a selected rule; include an arbitrary constant because differentiation cannot reveal the original constant term.

Formula

This calculator covers two basic indefinite integrals: the power rule, ∫a xⁿ dx = a xⁿ⁺¹/(n+1) + C, when n ≠ −1; and the logarithm rule, ∫a/x dx = a ln|x| + C. Here a is a dimensionless numerical coefficient, n is a dimensionless exponent, x is the variable of integration, and C is an arbitrary constant. The input x is understood to be the symbolic variable, not a numerical value. The calculator’s output is an expression, not a numerical measurement, so it has no physical unit unless x and the integrand represent quantities with units.

The rule selector indicates which pattern to apply: choose Power rule for a xⁿ, and Logarithm rule for a/x. For the power rule, the displayed formula applies when n is not −1. The logarithm case is the exceptional power n = −1, since substituting that exponent into the power-rule formula would require division by zero. The block provides a simple power-rule calculation; use the stated logarithm rule for the exceptional case.

Worked Example

With the defaults, a = 3, n = 2, and the power rule is selected. The integrand is therefore 3x². Increase the exponent by one, divide by the new exponent, and retain the coefficient:

• ∫3x² dx = 3 × x³/3 + C
• Cancel the factors of 3: x³ + C
• Differentiate to check: d(x³ + C)/dx = 3x²

The result is x³ + C. In the calculator expression, the same result is written as 3 * (x ^ 3) / 3, with “+ C” identifying the integration constant.

What an Indefinite Integral Represents

An indefinite integral is not one number. It represents all antiderivatives of a function on an interval. For example, x³, x³ + 5, and x³ − 12 all have derivative 3x²; they differ only by a constant. That is why the general answer includes C.

This contrasts with a definite integral, which has lower and upper limits and returns a number representing signed accumulated change. If a problem asks for total distance, area, or another accumulated quantity over a specified interval, it may require a definite integral rather than an indefinite one. The two are connected by the fundamental theorem of calculus: find an antiderivative, then evaluate its change between the endpoints.

How to Choose an Integration Rule

Start by identifying the form of the integrand. Basic rules work directly for familiar patterns; more complicated expressions often need rewriting or a method such as substitution or integration by parts. Useful first checks include:

• Power rule: For xⁿ, add 1 to the exponent and divide by the new exponent, provided n ≠ −1. For instance, ∫x⁴ dx = x⁵/5 + C.
• Constant multiple: A constant factor can be kept outside the integral: ∫7x² dx = 7∫x² dx = 7x³/3 + C.
• Sum and difference: Integrate each term separately: ∫(x² + 4x) dx = x³/3 + 2x² + C.
• Reciprocal: For x⁻¹ = 1/x, use ∫1/x dx = ln|x| + C, not the power rule.

For a product or a function nested inside another, do not apply the power rule mechanically. Substitution can simplify a composite expression when the derivative of its inner function is also present. Integration by parts is often useful for products such as x e^x. Trigonometric and exponential functions also have their own standard formulas.

Checking an Answer and Avoiding Mistakes

The most dependable check is to differentiate the proposed antiderivative. The derivative should reproduce the original integrand. This catches common errors such as forgetting to divide by the new exponent, changing a sign, or omitting a factor from the chain rule.

• Do not forget + C in a general indefinite integral. It is unnecessary only when a particular antiderivative is requested with a specified condition or convention.
• Check the exponent carefully: ∫x² dx is x³/3 + C, not x³ + C.
• For ∫1/x dx, write ln|x| + C. The absolute value accounts for intervals where x is negative; the integrand itself is undefined at x = 0.
• Keep expressions exact where possible. Rounding coefficients during intermediate steps can make a correct result appear inconsistent when differentiated.

Antiderivatives are generally considered on intervals where the integrand is defined and continuous. A function can have different constants on separate disconnected intervals, so the domain matters when interpreting a full family of solutions.

Limits of This Calculator

The power-rule calculator handles a coefficient times a single power of x. It does not automatically recognize algebraically equivalent forms, integrate sums term by term, or select advanced techniques. For example, ∫2x(x² + 1)⁴ dx requires recognizing a composite pattern; a basic power-rule input alone does not capture the whole expression.

Likewise, the formula cannot produce a single numerical value for an indefinite integral without additional information, such as a condition that determines C. If a function F must satisfy F(0) = 4, first find the general antiderivative and then use that condition to solve for C. Computer algebra systems can handle many more symbolic forms, but their output should still be checked by differentiation and interpreted with attention to domains.

Frequently Asked Questions

What does an indefinite integral give you?
It gives a family of functions whose derivative is the integrand. The family is represented by an antiderivative plus an arbitrary constant, C.

Why do you add C when integrating?
Because the derivative of every constant is zero. Differentiation erases constant terms, so integration cannot determine which one was present in the original function.

What is the integral of 1/x?
It is ln|x| + C on any interval that does not cross x = 0. The logarithm rule is the special case excluded from the ordinary power rule.

How can I check an indefinite integral?
Differentiate your answer. If the result matches the original integrand on the relevant domain, the antiderivative is correct.

Is an indefinite integral the same as area under a curve?
No. An indefinite integral is a family of antiderivatives. A definite integral over an interval gives a signed accumulation, which can represent area under suitable conditions.

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in a | Coefficient a | - | 3 | 
in n | Power n | - | 2 | 0
sel r | Rule | Power rule=1, Logarithm rule=2
out f | Indefinite integral | + C | a * (x ^ (n + 1)) / (n + 1) | 2