antiderivative calculator - Aaron Graves, PhDude Replica

Free Antiderivative Calculator

Enter a function and compute an indefinite integral instantly.

Function f(x)

Variable of integration

Try an example:

What is an antiderivative?

An antiderivative of a function f(x) is a function F(x) whose derivative is f(x). In symbols, if F'(x) = f(x), then F(x) is an antiderivative of f(x). This is also called an indefinite integral and is written as: ∫ f(x) dx = F(x) + C, where C is the constant of integration.

Antiderivatives are foundational in calculus, differential equations, physics, probability, and economics. Anytime you want to reverse differentiation, recover a position from velocity, or compute area accumulation, you are using integration.

How to use this antiderivative calculator

Type your expression in standard calculator/CAS style (for example: x^3 + 2*x).
Use function names such as sin(x), cos(x), ln(x), e^x, and sqrt(x).
Set the variable (usually x).
Click Calculate Antiderivative to compute the indefinite integral.
Remember to include + C in final written work.

Common integration rules (quick reference)

Power rule

For n ≠ -1: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C

Exponential and logarithmic forms

∫ e^x dx = e^x + C
∫ a^x dx = a^x/ln(a) + C, for a > 0 and a ≠ 1
∫ 1/x dx = ln|x| + C

Basic trigonometric antiderivatives

∫ sin(x) dx = −cos(x) + C
∫ cos(x) dx = sin(x) + C
∫ sec²(x) dx = tan(x) + C

Worked examples

Example 1: Polynomial

If f(x) = 6x^2 - 4x + 9, integrate term-by-term: ∫(6x^2 - 4x + 9)dx = 2x^3 - 2x^2 + 9x + C.

Example 2: Trig and exponential mix

For f(x)=cos(x)+e^x, apply known forms directly: ∫(cos(x)+e^x)dx = sin(x) + e^x + C.

Example 3: Substitution-style pattern

For (3x+1)^5, a substitution u=3x+1 gives: ∫(3x+1)^5 dx = (3x+1)^6 / 18 + C.

When a symbolic result is hard

Some integrals do not simplify into elementary functions. A symbolic engine might return an unevaluated integral, or an expression involving special functions. That is normal in advanced calculus. If this happens, you can:

Rewrite the input (factor, simplify, or separate terms).
Try substitution manually before entering the expression.
Use numerical integration for definite bounds when an elementary antiderivative is unavailable.

Study tips for mastering antiderivatives

Memorize core derivative-antiderivative pairs.
Practice linearity: integrate sums and constants efficiently.
Train pattern recognition for substitution and trig identities.
Always verify by differentiating your final answer.

FAQ

Does this calculator solve definite integrals?

This tool focuses on indefinite integrals (antiderivatives). For definite integrals, compute the antiderivative first, then apply upper and lower limits.

Why is “+ C” required?

Because the derivative of any constant is zero, infinitely many antiderivatives differ only by a constant.

What input format works best?

Use explicit multiplication like 3*x, powers as x^2, and function syntax like sin(x), ln(x), and sqrt(x).