Integration Calculator

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integration calculator
Indefinite Integral (Power Rule: axⁿ)Definite Integral (Bound [a, b])
Result:
Enter values to calculate
function toggleBounds(){var type=document.getElementById('given_data').value;var rowA=document.getElementById('boundRowA');var rowB=document.getElementById('boundRowB');if(type==='definite'){rowA.style.display='table-row-group';rowB.style.display='table-row-group';}else{rowA.style.display='none';rowB.style.display='none';}}function calculateResult(){var a=parseFloat(document.getElementById('coeff').value);var n=parseFloat(document.getElementById('power').value);var type=document.getElementById('given_data').value;var showSteps=document.getElementById('steps').checked;var resultVal=document.getElementById('resultValue');var stepsDisp=document.getElementById('stepsDisplay');if(isNaN(a)||isNaN(n)){alert('Please enter valid numerical values for coefficient and power.');return;}if(n===-1){resultVal.innerHTML="∫("+a+"x⁻¹) dx = "+a+" * ln|x| + C";if(showSteps){stepsDisp.style.display='block';stepsDisp.innerHTML="Using rule: ∫(1/x)dx = ln|x|. Result is logarithmic.";}return;}var newPower=n+1;var newCoeff=a/newPower;if(type==='indefinite'){var resString="∫("+a+"x"+n+") dx = ";if(newCoeff===1){resString+="x"+newPower+" + C";}else if(newCoeff===-1){resString+="-x"+newPower+" + C";}else{resString+=newCoeff.toFixed(4).replace(/\.?0+$/,"")+"x"+newPower+" + C";}resultVal.innerHTML=resString;if(showSteps){stepsDisp.style.display='block';stepsDisp.innerHTML="Step 1: Add 1 to power: "+n+" + 1 = "+newPower+"
Step 2: Divide coefficient by new power: "+a+" / "+newPower+" = "+newCoeff.toFixed(4)+"
Step 3: Add constant of integration C.";}}else{var low=parseFloat(document.getElementById('lower').value);var up=parseFloat(document.getElementById('upper').value);if(isNaN(low)||isNaN(up)){alert('Please enter valid bounds.');return;}var valUp=newCoeff*Math.pow(up,newPower);var valLow=newCoeff*Math.pow(low,newPower);var finalVal=valUp-valLow;resultVal.innerHTML="Result = "+finalVal.toFixed(4).replace(/\.?0+$/,"");if(showSteps){stepsDisp.style.display='block';stepsDisp.innerHTML="Step 1: Find antiderivative: F(x) = ("+a+"/"+newPower+")x"+newPower+"
Step 2: Evaluate at Upper Bound ("+up+"): "+valUp.toFixed(4)+"
Step 3: Evaluate at Lower Bound ("+low+"): "+valLow.toFixed(4)+"
Step 4: Subtract: "+valUp.toFixed(4)+" – "+valLow.toFixed(4)+" = "+finalVal.toFixed(4);}}}function resetCalculator(){document.getElementById('stepsDisplay').style.display='none';document.getElementById('resultValue').innerHTML='Enter values to calculate';toggleBounds();}

Integration Calculator Use

The integration calculator is a specialized mathematical tool designed to find the antiderivative of functions and evaluate definite integrals. Whether you are a student solving calculus homework or an engineer calculating areas under a curve, this calculator provides instant solutions using the fundamental power rule of integration.

To use this tool effectively, select your calculation type (indefinite or definite) and input the coefficient and power of your variable. The calculator handles positive, negative, and fractional exponents, making it versatile for various mathematical applications.

Coefficient (a)
The numerical value multiplying the variable (e.g., in 5x², the coefficient is 5).
Power (n)
The exponent the variable is raised to. Note that if n = -1, the result becomes a natural logarithm.
Lower/Upper Bounds
Used for definite integrals to find the exact numerical area between two points on the x-axis.

How Integration Works

Integration is the reverse process of differentiation. When using the integration calculator for power functions, we apply the Power Rule for Integration. This rule is essential for basic calculus and forms the foundation for more complex techniques.

∫ axⁿ dx = (a / (n+1)) * xⁿ⁺¹ + C

  • a: The constant coefficient.
  • n: The original power of the variable.
  • n+1: The new power after integration.
  • C: The constant of integration (required for indefinite integrals).

Indefinite vs. Definite Integrals

Understanding the difference between these two types of calculations is key to using the integration calculator correctly:

Indefinite Integrals

An indefinite integral represents a family of functions. Because the derivative of a constant is zero, when we reverse the process, we must add a "+ C" to represent any possible constant that might have existed in the original function. The output is always a functional expression.

Definite Integrals

A definite integral calculates the signed area between the function and the x-axis over a specific interval [a, b]. It yields a specific numerical value. To find this, we calculate the antiderivative at the upper bound and subtract the antiderivative value at the lower bound.

Calculation Example

Example: Evaluate the definite integral of 3x² from x = 1 to x = 3.

Step-by-step solution:

  1. Identify components: a = 3, n = 2.
  2. Find the antiderivative: F(x) = (3 / (2+1))x²⁺¹ = 3/3 x³ = x³.
  3. Evaluate at upper bound (3): 3³ = 27.
  4. Evaluate at lower bound (1): 1³ = 1.
  5. Subtract the results: 27 – 1 = 26.
  6. Final Result = 26

Common Questions

What happens if the power is -1?

If you try to apply the standard power rule to x⁻¹, you would end up dividing by zero (-1 + 1 = 0). Therefore, a special rule is used: the integral of 1/x is the natural logarithm, ln|x| + C. The integration calculator automatically detects this case.

What is the "C" in the results?

"C" stands for the constant of integration. It represents an arbitrary constant because many different functions (like x²+5, x²+10, or x²-100) all have the same derivative (2x). When integrating 2x, we include C to account for any potential vertical shift.

Can this calculator handle square roots?

Yes. To integrate a square root like √x, you should convert it to an exponent first. Since √x = x⁰⁵, you would enter "1" as the coefficient and "0.5" as the power in the integration calculator.

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