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Computer algebra systems sometimes need a helping hand from human beings. Try to evaluate$$ \int (1 + \ln x) \sqrt{1 + (x \ln x)^2}\ dx $$with a computer algebra system. If it doesn't return an answer, make a substitution that changes the integral into one that the $ CAS $ can evaluate.

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 6

Integration Using Tables and Computer Algebra Systems

Integration Techniques

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

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.

00:46

Computer algebra systems s…

02:51

Machines sometimes need a …

04:29

Attempt to use a computer …

00:25

Use a computer algebra sys…

05:18

Use an appropriate substit…

01:16

There are often multiple w…

00:49

06:38

Some integrals that can be…

Okay. This question wants us to evaluate the following integral using technology. Sometimes if your computer algebra system is older or just depending on the manufacturer, sometimes it can't do this out. Right. So we need to help it a little bit by putting in one of the forms or tables? No. So let's just rewrite it a little smaller here. So we see what we're working with. So most of our integral table forms have some sort of a you squared inner radical. So let's let you equal X Ellen X. Okay, so then d'you is using the product rule the derivative of X times, Ellen X. So Alan X plus x times the derivative of Ellen Ex So X, divided by x no DX. So this means that D. U is equal to Alan of X plus one D. X up and look at that. We have that in our integral, So this becomes the integral of square root one plus, Do you squared, do you? And there's a very simple form that we can just plug into an integral calculator or, if you really want we've been practicing with tables, so you could just find this in the table. And either way, this should give you you over two times the square root of one plus u squared plus 1/2 times inverse, hyperbolic sign of you plus e So again, the only questionable thing is some computer Alger systems like mine returned. Hyperbolic sign, inverse of you squared. But others may use the logger of them definition of inverse hyperbolic sign. So now when we have to do is substitute x l on X back in for you and get excel in X over too square root one plus x Ln x squared, plus 1/2 times inverse, hyperbolic sign of ex Allen X squared plus c So this is our final answer. So again, sometimes if you have a really complicated looking integral, what you're gonna want to d'oh is converted into a form that is likely to be recognizable by your computer algebra system.

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