Integral Calculator — Antiderivative with Step-by-Step
Calculate indefinite integrals of polynomial, trigonometric, and exponential functions. Get the antiderivative with +C and step-by-step integration rules.
Integrand f(x)
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Antiderivative F(x) + C
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Step-by-Step Solution
Integration Rules Reference
- Power Rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
- Constant Rule: ∫k dx = kx + C
- Sin Rule: ∫sin(x) dx = -cos(x) + C
- Cos Rule: ∫cos(x) dx = sin(x) + C
- Exponential Rule: ∫eˣ dx = eˣ + C
- Log Rule: ∫(1/x) dx = ln|x| + C
- Ln Rule: ∫ln(x) dx = x·ln(x) - x + C
Frequently Asked Questions
What is an integral?
An integral is the reverse process of differentiation. The indefinite integral of f(x) gives a family of functions F(x) + C such that F'(x) = f(x). The constant C represents any real number and is called the constant of integration.
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.
What is the integral of sin(x)?
∫sin(x) dx = -cos(x) + C. More generally, ∫sin(bx) dx = -cos(bx)/b + C using u-substitution.
What is the integral of eˣ?
∫eˣ dx = eˣ + C. For ∫eᵃˣ dx = eᵃˣ/a + C using u-substitution where u = ax.
Why do we add +C to indefinite integrals?
The +C (constant of integration) accounts for the fact that the derivative of any constant is zero. So when reversing differentiation, we cannot determine any constant term, meaning infinitely many antiderivatives exist differing only by a constant.