find indefinite integral calculator

What this calculator does

This page helps you quickly find an indefinite integral, also called an antiderivative. If you provide a function such as x^2, the tool returns a function whose derivative is that input. It appends the constant of integration (+ C) because infinitely many antiderivatives differ by a constant.

How to use the indefinite integral calculator

1) Type the integrand

Enter the expression using standard calculator-style syntax. Use explicit multiplication: write 3*x instead of 3x.

2) Choose the variable

By default, integration is with respect to x. You can switch to another symbol such as t or y if your expression uses a different variable.

3) Click “Calculate Integral”

The result appears in symbolic form, followed by + C. The calculator also performs a derivative check in the background when possible.

Input syntax guide

• Powers: x^5, (x+1)^2
• Multiplication: 2*x, x*sin(x)
• Trig: sin(x), cos(x), tan(x)
• Exponential: exp(x) or e^x
• Logarithm: log(x) (natural log)
• Fractions: 1/(1+x^2), (x^2+1)/x

Why the “+ C” is always required

If F'(x) = f(x), then (F(x) + k)' = f(x) for any constant k. That means antiderivatives are not unique. The + C records this family of solutions. You only get a single specific function when an initial condition is provided, as in an initial value problem.

Common results you should recognize

• ∫ x^n dx = x^(n+1)/(n+1) + C for n != -1
• ∫ 1/x dx = log(x) + C
• ∫ sin(x) dx = -cos(x) + C
• ∫ cos(x) dx = sin(x) + C
• ∫ exp(ax) dx = exp(ax)/a + C for constant a

Good habits when checking your answer

Even with a symbolic calculator, you should verify quickly:

• Differentiate the returned antiderivative and compare it to your original input.
• Watch for domain restrictions (especially logs and square roots).
• Simplify algebraically before integrating when possible.
• Use parentheses carefully to avoid precedence mistakes.

FAQ

Does this solve every possible integral?

No symbolic engine can produce elementary closed forms for every function. If no closed form is found, try rewriting the expression or using a numerical method for definite integrals.

Can I use this for learning?

Yes. It is useful for homework checks, practice drills, and quick verification while studying calculus. For best learning outcomes, attempt the integration manually first, then compare.

Is this a definite integral calculator?

This page focuses on indefinite integrals. It returns antiderivatives, not area values over bounds.