Indefinite Integral Calculator

Enter any function of x into the Indefinite Integral Calculator and get the antiderivative (indefinite integral) returned with the constant of integration + C. Supports polynomials, trigonometric, exponential, logarithmic, and rational functions. Choose your integration variable, select a function type preset, and the calculator computes ∫f(x)dx for you.

Select the type of function you want to integrate.

The variable with respect to which you integrate.

Leading coefficient of your function term.

The power for polynomial or root types (e.g. n=2 gives x²). For trig/exp/log, this multiplies the argument.

Coefficient inside the argument, e.g. b in sin(bx), e^(bx), or 1/(bx+c).

Constant added inside the argument, e.g. c in (bx + c)ⁿ or 1/(bx + c).

Display which integration rule was applied.

Results

Antiderivative F(x) + C

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f(x) — Original Function

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Integration Rule Applied

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Verification (d/dx of result)

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f(x) vs F(x) — Function and Its Antiderivative

Results Table

Frequently Asked Questions

What is an indefinite integral?

An indefinite integral (antiderivative) is the reverse operation of differentiation. For a function f(x), the indefinite integral ∫f(x)dx finds a family of functions F(x) + C such that F'(x) = f(x). The constant C represents an arbitrary constant of integration, since the derivative of any constant is zero.

Why does the result always include '+ C'?

The constant of integration C appears because differentiation eliminates any constant. For example, both x² + 5 and x² − 3 have derivative 2x, so ∫2x dx = x² + C covers all possible antiderivatives at once. When you have boundary conditions or initial values, C can be determined.

What is the Power Rule for integration?

The Power Rule states that ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, provided n ≠ −1. For example, ∫x³ dx = x⁴/4 + C. When n = −1, the integral is ∫(1/x) dx = ln|x| + C.

How do I integrate trigonometric functions?

Common trig integrals include: ∫sin(x) dx = −cos(x) + C, ∫cos(x) dx = sin(x) + C, and ∫tan(x) dx = −ln|cos(x)| + C. With a coefficient b inside the argument (e.g. sin(bx)), divide by b: ∫sin(bx) dx = −cos(bx)/b + C.

What is the integral of eˣ and other exponentials?

The exponential function is its own antiderivative: ∫eˣ dx = eˣ + C. With a coefficient, ∫e^(bx) dx = e^(bx)/b + C. For other bases, ∫aˣ dx = aˣ / ln(a) + C where a > 0, a ≠ 1.

What techniques are used to solve complex integrals?

The most common integration techniques include: the Power Rule (for polynomials), u-Substitution (for composite functions), Integration by Parts (∫u dv = uv − ∫v du, for products), Partial Fractions (for rational functions), and Trigonometric Substitution (for expressions involving √(a²−x²) and similar forms).

How can I verify my antiderivative is correct?

To verify an antiderivative F(x), simply differentiate it. If F'(x) = f(x), the answer is correct. For example, if you claim ∫3x² dx = x³ + C, check: d/dx(x³ + C) = 3x² ✓. This calculator's verification row shows this check for supported function types.

What is the difference between a definite and indefinite integral?

An indefinite integral ∫f(x)dx produces a general antiderivative F(x) + C with no numerical value. A definite integral ∫[a to b] f(x) dx uses the Fundamental Theorem of Calculus — it equals F(b) − F(a) and gives a specific number representing the net area under the curve between x = a and x = b.

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