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Problem 46 Hard Difficulty

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.

Answer

$\frac{x \ln x \sqrt{1+(x \ln x)^{2}}}{2}+\frac{1}{2} \arcsin h\left[(x \ln x)^{2}\right]$

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Video Transcript

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.