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Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.
$ \displaystyle \int x^2 \sqrt{x^2 + 4}\ dx $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 6
Integration Using Tables and Computer Algebra Systems
Integration Techniques
Harvey Mudd College
Baylor University
University of Michigan - Ann Arbor
Lectures
01:53
In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.
27:53
In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.
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Use a computer algebra sys…
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Use a computer algebra sy…
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Okay, so this question wants us to evaluate this anti derivative using a computer algebra system and show us that that's the same thing that we get if we use the table. So I plugged into a computer algebra system and got this answer. But when you plug into a table, you get this form. So let's show they're the same, but converting the computer form into the tabular for So let's distribute this 1/4 to each term. So we get square root X squared, plus four over four times Ex cued plus two X minus eight over for Ellen of X plus Square root X squared plus four plus C. Then we can factor out X from the first term. So we get X over four times X squared plus two X minus. Sorry. Need our square roots still minus two. Ellen of X was squared, X squared, plus floor plus c. And then this looks really close to our table for him. We just need to pull out a factor of 1/2 from the first term. So we get X over eight times. Well, if we pulled out a factor on half if to multiply, both things and the parentheses by two, and we leave the second term alone, and this should be our final answer. So let's compare this. We get X over a times two X squared plus four that are square term. Yep, Then minus to Ellen of X plus Squared X squared plus four, which we do indeed get, making sure to expand just for clarity and the answer's air equivalent.
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