Antiderivative Calculator with Steps
Antiderivative F(x):
Understanding Antiderivatives and How to Calculate Them
An antiderivative, also known as an indefinite integral, is the reverse process of differentiation. If you have a function f(x), its antiderivative F(x) is a function such that the derivative of F(x) is f(x). In simpler terms, if you differentiate F(x), you get back f(x).
The Constant of Integration (+ C)
When finding an antiderivative, there's always an arbitrary constant 'C' added to the result. This is because the derivative of any constant is zero. For example, the derivative of x² is 2x, and the derivative of x² + 5 is also 2x. So, when integrating 2x, the antiderivative could be x², x² + 5, x² – 10, or any x² + C. This 'C' represents all possible constant terms.
Why are Antiderivatives Important?
Antiderivatives are fundamental in calculus and have wide-ranging applications:
- Physics: If you know the acceleration of an object (which is the derivative of velocity), you can find its velocity by taking the antiderivative. Similarly, integrating velocity gives you position.
- Engineering: Used in structural analysis, fluid dynamics, and electrical engineering to model and solve problems involving rates of change.
- Economics: Can be used to find total cost or total revenue functions from marginal cost or marginal revenue functions.
- Probability: Essential for calculating probabilities from probability density functions.
- Area Under a Curve: The definite integral, which is based on antiderivatives, is used to calculate the area under a curve, volumes, and other geometric properties.
Basic Antiderivative Rules
Here are some common rules for finding antiderivatives:
- Power Rule: For a term like \(ax^n\), the antiderivative is \(\frac{a}{n+1}x^{n+1}\), provided \(n \neq -1\).
- Constant Rule: For a constant \(a\), the antiderivative is \(ax\).
- Reciprocal Rule: For \(1/x\), the antiderivative is \(\ln|x|\).
- Exponential Rule: For \(e^x\), the antiderivative is \(e^x\).
- Trigonometric Rules:
- \(\int \sin(x) dx = -\cos(x)\)
- \(\int \cos(x) dx = \sin(x)\)
- Sum/Difference Rule: The antiderivative of a sum or difference of functions is the sum or difference of their antiderivatives.
How to Use This Calculator
Our Antiderivative Calculator simplifies the process of finding indefinite integrals for common functions. Simply enter your function f(x) into the input field. The calculator supports:
- Polynomial terms (e.g.,
x^2,3x,-5) - Reciprocal terms (e.g.,
1/x,-2/x) - Exponential terms (e.g.,
e^x,4e^x) - Basic trigonometric functions (e.g.,
sin(x),cos(x))
The calculator will then display the antiderivative F(x) and provide step-by-step details for each term, making it a great tool for learning and checking your work.
Examples:
- Input:
3x^2 + 4x - 5
Output:F(x) = x^3 + 2x^2 - 5x + C - Input:
sin(x) + e^x
Output:F(x) = -cos(x) + e^x + C - Input:
1/x + 6x^3
Output:F(x) = ln|x| + 1.5x^4 + C
" + remainingInput + ". Please check your input format.";
stepsDiv.innerHTML = "";
return;
}
}
// Format the final antiderivative
var finalAntiderivative = "";
for (var k = 0; k 0 && !term.startsWith("-")) {
finalAntiderivative += " + ";
} else if (k > 0 && term.startsWith("-")) {
finalAntiderivative += " "; // Space before negative term
}
finalAntiderivative += term;
}
if (finalAntiderivative === "") {
finalAntiderivative = "0"; // If all terms integrated to 0 (e.g., input was "0")
}
finalAntiderivative += " + C";
antiderivativeResultDiv.innerHTML = "F(x) = " + finalAntiderivative;
stepsDiv.innerHTML = "Steps:
- ";
for (var l = 0; l < stepDetails.length; l++) {
stepsDiv.innerHTML += "
- " + stepDetails[l] + " "; } stepsDiv.innerHTML += "