Integral of (2x^2+3)^2: Solution

In summary, the basic formula for finding the integral of (2x^2+3)^2 is ∫(2x^2+3)^2 dx = ∫4x^4+12x^2+9 dx = (4/5)x^5 + 4x^3 + 9x + C, where C is the constant of integration. This integral can also be solved using substitution, where u = 2x^2+3 and du/dx = 4x. Using the trapezoidal rule, we can approximate the integral by dividing the area under the curve into trapezoids. The geometric interpretation of finding this integral is the area under the curve of the function. A
  • #1
integral (2x^2+3)^2dx=?
 
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  • #2
Expand that polynomial out first, then you can integrate it termwise easily.
 

1. What is the basic formula for finding the integral of (2x^2+3)^2?

The basic formula for finding the integral of (2x^2+3)^2 is ∫(2x^2+3)^2 dx = ∫4x^4+12x^2+9 dx = (4/5)x^5 + 4x^3 + 9x + C, where C is the constant of integration.

2. How do you solve the integral of (2x^2+3)^2 using substitution?

To solve the integral of (2x^2+3)^2 using substitution, let u = 2x^2+3. Then du/dx = 4x and dx = du/4x. Substituting these values into the integral, we get ∫(2x^2+3)^2 dx = ∫u^2 (du/4x) = (1/4)∫u^2 du. This can be easily integrated using the power rule, giving us the final solution of (1/4)(u^3/3) + C = (1/12)(2x^2+3)^3 + C.

3. Can the integral of (2x^2+3)^2 be solved using the trapezoidal rule?

Yes, the integral of (2x^2+3)^2 can be solved using the trapezoidal rule. This numerical integration method approximates the integral by dividing the area under the curve into trapezoids and summing their areas. The more trapezoids used, the more accurate the approximation will be.

4. What is the geometric interpretation of finding the integral of (2x^2+3)^2?

The geometric interpretation of finding the integral of (2x^2+3)^2 is the area under the curve of the function. This can be visualized by graphing the function and shading the area under the curve between the limits of integration. This area represents the total change or accumulation of the function over the given interval.

5. Is there a shortcut or easier way to solve the integral of (2x^2+3)^2?

Yes, there is a shortcut or easier way to solve the integral of (2x^2+3)^2. Instead of using substitution, we can use the power rule for integration, which states that ∫x^n dx = (1/(n+1))x^(n+1) + C. Applying this rule to our integral, we get (1/5)(2x^2+3)^5 + C as the solution.

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