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Problem 43 Medium Difficulty

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 \tan^5 x\ dx $

Answer

$$\frac{1}{4} \tan ^{4} x-\frac{1}{2} \tan ^{2} x-\ln |\cos x|+C$$

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Video Transcript

Okay, So this question wants us to evaluate this integral using technology and show that its form is equivalent to that given by the table. So a computer algebra system gives us this anti derivative. Well, the table gives us this one says you can see the tangent to the fourth term and the change into the second term both match. So we just need to show that these 2/3 terms are equivalent. So I need to show that negative Ellen of C can't axe equals hello and of co sign X. Okay, So this is pretty straightforward to dio because consider negative alone of seeking ex. So is the same thing is negative, Ellen of well, c can't is one over co sign X. So that means it's negative. Ellen of co sign X all raised to the negative first power end by lug properties. We can bring this exponents outfront and get negative one times negative Ln of coastline X and that simplifies to just Ellen of co sign X. So we've shown that negative Ellen of C Can X is the same thing. Is Ellen of co sign X So we're done because that means that our table and computer algebra system answers are absolutely the same

University of Michigan - Ann Arbor
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