Integral Calculator
Find the antiderivative/integral of any function with step-by-step explanations and interactive graphs.
Our Integral Calculator helps you find the antiderivative (indefinite integral) of functions effortlessly. It provides detailed step-by-step explanations of the integration rules applied, such as the Power Rule, Sum Rule, and Substitution, making it an excellent tool for learning calculus.
Visualize the relationship between the function f(x) and its integral F(x) with our interactive grapher.
Enter Function f(x)
Graph Evaluation
Computed Integral
What is Integral Calculator?
An Integral (specifically an indefinite integral) represents the reverse process of differentiation. If differentiation finds the slope of a curve, integration finds the area under the curve (for definite integrals) or the original function (for indefinite integrals).
The indefinite integral of a function f(x) is written as:
Where F(x) is the antiderivative (such that F'(x) = f(x)) andC is the constant of integration.
Formula & Calculation
- Power Rule:∫ x^n dx = (x^(n+1))/(n+1) + C (for n ≠ -1)
- Constant Multiple Rule:∫ c·f(x) dx = c · ∫ f(x) dx
- Sum/Difference Rule:∫ (f(x) ± g(x)) dx = ∫ f(x) dx ± ∫ g(x) dx
- Common Integrals:∫ sin(x) dx = -cos(x) + C
∫ cos(x) dx = sin(x) + C
∫ e^x dx = e^x + C
∫ (1/x) dx = ln|x| + C
Example Calculation
Example: Find the integral of f(x) = 3x^2 + 2x.
- Apply the Sum Rule: ∫ (3x^2 + 2x) dx = ∫ 3x^2 dx + ∫ 2x dx
- Apply Constant Multiple Rule: 3∫ x^2 dx + 2∫ x dx
- Apply Power Rule to each term:
- ∫ x^2 dx = x^3/3
- ∫ x^1 dx = x^2/2
- Combine and Simplify: 3(x^3/3) + 2(x^2/2) = x^3 + x^2 + C
Frequently Asked Questions
What is 'C' in the answer?
The 'C' stands for the Constant of Integration. Because the derivative of any constant is zero, there are infinitely many antiderivatives for a function, differing only by a constant value. We add 'C' to represent this family of functions.
Can this calculator solve definite integrals?
Currently, this calculator focuses on indefinite integrals (antiderivatives). However, you can use the result to calculate a definite integral by evaluating F(b) - F(a) manually.
Why can't some functions be integrated?
Not all functions have an antiderivative that can be expressed in terms of elementary functions (like polynomials, exponentials, and trig functions). These are called "non-elementary integrals" (e.g., e^(-x^2)).