int_{frac{1}{2}}^{2} frac{pi xleft(4 x^{2}+3right) sqrt{left(4 x^{2}+1right)^{2}+16}}{24} d x

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Step:1 ：We are given the definite integral problem: (int_{frac{1}{2}}^{2} frac{pi xleft(4 x^{2}+3right) sqrt{left(4 x^{2}+1right)^{2}+16}}{24} d x)Step:2 ：To solve this, we can use numerical integration methods. We first define the function inside the integral.Step:3 ：Then, we calculate the integral from 1/2 to 2.Step:4 ：The result of the calculation is approximately 33.71361521007593.Step:5 ：Final Answer: The definite integral of the given function from 1/2 to 2 is (boxed{33.71361521007593})

Answer

Step: 1 ：We are given the definite integral problem: (int_{frac{1}{2}}^{2} frac{pi xleft(4 x^{2}+3right) sqrt{left(4 x^{2}+1right)^{2}+16}}{24} d x)Step: 2 ：To solve this, we can use numerical integration methods. We first define the function inside the integral.Step: 3 ：Then, we calculate the integral from 1/2 to 2.Step: 4 ：The result of the calculation is approximately 33.71361521007593.Step: 5 ：Final Answer: The definite integral of the given function from 1/2 to 2 is (boxed{33.71361521007593})

$\int_{\frac{1}{2}}^{2} \frac{\pi x\left(4 x^{2}+3\right) \sqrt{\left(4 x^{2}+1\right)^{2}+16}}{24} d x$

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