High School Indefinite Integral Calculator
Visualize the family of antiderivatives F(x) + C and slope fields.
Enter function to integrate (e.g., 2x, cos(x), x^2)
Calculated Result
*Symbolic support is limited to basic polynomials and trig.
Slope Field & Family of Curves
Understanding the Indefinite Integral Calculator
An indefinite integral, usually written as ∫ f(x) dx, represents the general antiderivative of a function. Unlike a definite integral, which calculates a specific area value, an indefinite integral produces a function (or a family of functions).
The Family of Curves & Constant ‘C’
If F(x) is an antiderivative of f(x), then F'(x) = f(x). However, the derivative of any constant number is 0. This means that F(x) + 5, F(x) – 100, and F(x) + Ï€ all have the same derivative.
Therefore, the general solution is always written as F(x) + C, where C is the arbitrary constant of integration. Visually, this creates a “family” of parallel curves, all vertically shifted versions of each other.
Integration by Substitution
Often called “Reverse Chain Rule,” substitution is a method to simplify complex integrals.
Steps:
- Choose a part of the integrand to be u (usually the “inner” function).
- Find du/dx and solve for dx.
- Substitute u and dx into the integral. All x terms must disappear.
- Integrate with respect to u.
- Substitute back u in terms of x.
Why use an Indefinite Integral Calculator?
Visualizing antiderivatives is difficult because we are conditioned to think of functions as single lines. This tool generates a Slope Field, where small line segments represent the slope f(x) at every point. The integral curves simply “flow” through these slopes, providing a beautiful geometric intuition for what integration actually does.
FAQ
Yes, this tool is completely free and runs in your browser using numerical methods for graphing.
The multiple yellow lines represent the “Family of Curves” F(x) + C for different values of C. They all share the same slope at any given x-value.
While it primarily visualizes the result, the symbolic engine can recognize basic polynomial and trigonometric forms often found in introductory substitution problems.
A slope field visualizes a differential equation dy/dx = f(x). Each segment shows the direction the curve must travel at that point.