Antiderivative Calculator

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Antiderivative Calculator
Power Rule (ax^n)Linear Function (ax + b)Constant (c)
Indefinite Integral F(x):

Enter values and click Calculate

function calculateAntiderivative(){var mode=document.getElementById('calc_type').value;var a=parseFloat(document.getElementById('coeff_a').value);var n=parseFloat(document.getElementById('power_n').value);var b=parseFloat(document.getElementById('const_c').value);var plusC=document.getElementById('showSteps').checked?" + C":"";var result="";if(isNaN(a)){alert('Please enter a valid coefficient');return;}if(mode==="power"){if(isNaN(n)){alert('Please enter a valid power');return;}if(n===-1){result=a+" ln|x|"+plusC;}else{var newPower=n+1;var newCoeff=a/newPower;if(newCoeff===1){newCoeff="";}else if(newCoeff===-1){newCoeff="-";}else{newCoeff=Number(newCoeff.toFixed(4));}result=newCoeff+"x^"+newPower+plusC;}}else if(mode==="linear"){if(isNaN(b)){alert('Please enter a valid constant b');return;}var term1=a/2;var term2=b;result=(term1===1?"":term1===0?"":term1)+"x^2 + "+(term2===0?"":term2+"x")+plusC;}else if(mode==="constant"){result=a+"x"+plusC;}document.getElementById('resultValue').innerHTML="∫ f(x)dx = "+result;}

How to Use the Antiderivative Calculator

The antiderivative calculator is an essential tool for students and professionals working with calculus. It allows you to find the indefinite integral of functions quickly and accurately. Whether you are dealing with basic power rules or linear expressions, this tool handles the heavy lifting of algebraic manipulation.

To use this tool effectively, follow these steps:

Calculation Mode
Select the type of function you are integrating (Power Rule, Linear, or Constant).
Coefficient (a)
Enter the numerical multiplier in front of your variable.
Power (n)
Enter the exponent of the variable x. For example, in x^3, n would be 3.

How It Works: The Power Rule for Integration

Integration is the reverse process of differentiation. When using our antiderivative calculator, we primarily rely on the Fundamental Power Rule. The formula is expressed as:

∫ ax^n dx = (a / (n + 1)) * x^(n+1) + C

Here is what each component of the formula represents:

  • a: The coefficient of the term.
  • n: The current exponent of x (must not be -1 for the standard power rule).
  • n + 1: The new exponent after integration.
  • C: The constant of integration, representing any possible constant value that disappears during differentiation.

Calculation Examples

Example 1: Finding the antiderivative of 3x^2.

Using the antiderivative calculator for the function f(x) = 3x^2:

  1. Coefficient (a) = 3
  2. Power (n) = 2
  3. Add 1 to the power: 2 + 1 = 3
  4. Divide coefficient by new power: 3 / 3 = 1
  5. Result: 1x^3 + C or simply x^3 + C

Example 2: The special case of x^-1.

If you input a power of -1, the standard power rule results in division by zero. In this case, the antiderivative calculator applies the logarithmic rule: ∫ (1/x) dx = ln|x| + C.

Common Questions

What is a constant of integration (+ C)?

When you take the derivative of a constant (like 5 or 100), the result is zero. Therefore, when reversing the process via integration, we cannot know what the original constant was without more information. We use "+ C" to represent this unknown value.

Can this calculator handle trigonometric functions?

This specific version of the antiderivative calculator is optimized for algebraic power functions and polynomials. For trigonometric integrals (like sin or cos), advanced symbolic solvers are required, though the basic logic of reversing the derivative remains the same.

Why is the power rule different for n = -1?

Mathematically, if you apply the power rule to x^-1, you would get x^0 / 0. Since division by zero is undefined, calculus defines the antiderivative of 1/x specifically as the natural logarithm function, ln|x|.

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