Calculate The Integral 4x264 X2 Dx
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Reviewed by Calculator Editorial Team
This guide explains how to calculate the definite integral of 4x² + 64x² dx from a to b. We'll cover the formula, step-by-step calculation, and practical applications of this mathematical operation.
How to Calculate the Integral
Calculating the integral of 4x² + 64x² dx involves several steps. First, simplify the integrand by combining like terms. Then apply the power rule for integration to each term. Finally, evaluate the antiderivative at the upper and lower limits to find the definite integral.
Key Steps
- Combine like terms in the integrand
- Apply the power rule for integration to each term
- Evaluate the antiderivative at the limits of integration
- Subtract the lower limit evaluation from the upper limit evaluation
The power rule for integration states that for any real number n ≠ -1, the integral of xⁿ dx is (x^(n+1))/(n+1) + C, where C is the constant of integration. For definite integrals, we evaluate this antiderivative at the upper and lower limits and subtract.
Formula Used
The general formula for the definite integral of a function f(x) from a to b is:
Where F(x) is the antiderivative of f(x)
For our specific problem, the integrand is 4x² + 64x². We first combine these terms to get 68x², then apply the power rule to find the antiderivative.
Worked Example
Let's calculate the integral from 0 to 2 of 4x² + 64x² dx.
Step 1: Combine like terms
4x² + 64x² = 68x²Step 2: Apply the power rule
∫68x² dx = 68 * (x³/3) + C = (68/3)x³ + CStep 3: Evaluate at limits
[(68/3)(2)³] - [(68/3)(0)³] = (68/3)(8) - 0 = 544/3 ≈ 181.333...The result is 544/3, which is approximately 181.333 in decimal form.
Frequently Asked Questions
What is the integral of 4x² + 64x²?
The integral of 4x² + 64x² is (68/3)x³ + C, where C is the constant of integration.
How do I calculate a definite integral?
To calculate a definite integral, find the antiderivative of the integrand, evaluate it at the upper and lower limits, and subtract the lower evaluation from the upper evaluation.
What is the power rule for integration?
The power rule states that the integral of xⁿ dx is (x^(n+1))/(n+1) + C for any real number n ≠ -1.
Can I use this calculator for other integrals?
This calculator is specifically designed for integrals of the form 4x² + 64x² dx. For other integrals, you may need a different calculator or tool.