Integral Calculator
Compute indefinite integrals (antiderivatives) and definite integrals step-by-step with formulas, substitution, and + C shown.
Awaiting Input
Enter any equation like 2x + 8 = 20 or x² - 5x + 6 = 0 to get step-by-step solutions.
Understanding Integration
Integration is the inverse operation of differentiation. An indefinite integral represents a family of functions whose derivative equals the integrand.
Frequently Asked Questions
Why is the constant + C added to integrals?
For indefinite integrals, any constant term differentiates to zero. Therefore, there are infinitely many antiderivatives differing by a constant C, which represents this unknown constant.
What is the reverse power rule for integration?
The power rule for integration is ∫ x^n dx = (x^(n + 1)) / (n + 1) + C (for n ≠ -1). If n = -1, the integral is ln|x| + C.
Can this solve definite integrals with bounds?
Yes, enter limits to compute definite integrals using the Fundamental Theorem of Calculus: F(b) - F(a).
What is the difference between an indefinite integral and a definite integral?
An indefinite integral ∫ f(x) dx has no limits and represents the general family of antiderivatives, always including the constant of integration + C. A definite integral ∫[a to b] f(x) dx has specific integration bounds [a, b] and calculates a single numerical value representing the net signed area under the curve.
How do you solve integrals using u-substitution step by step?
First, identify an inner function u = g(x) whose derivative du = g'(x) dx also appears in the integrand. Next, substitute u and du to rewrite the integral completely in terms of u. Evaluate the simpler integral, and finally replace u with the original variable expression g(x) (adding + C for indefinite integrals).