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Problem 23 Medium Difficulty

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \sec^5 x\ dx $


$$\frac{1}{4} \tan x \sec ^{3} x+\frac{3}{8} \tan x \sec x+\frac{3}{8} \ln |\sec x+\tan x|+C$$


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Video Transcript

Okay, This question wants us to evaluate the integral of C can't x to the fifth power. So to do this, we consult our formula for seeking to the end anti derivative. So all we gotta do is apply the formula. So if an equals five says that this is equal to tangent X c can't to the and minus two, divided by n minus one plus and minus two over and minus one times the integral of C can't to the n minus two d x. And then we can use the same formula again to analyze, seek and cubed vex. So in a girl of C can't cubed iss tan axe c can't of n minus two, which is just the first power over and minus one plus and minus two over and minus one times the integral of C can't to the n minus two. And we actually know the internal of C can't to the first power, which is Ellen of Seek an X plus Tangent acts. So now let's back substitute everything in. So our final integral of seek and to the fifth I'm just gonna do a little box over here because we need a lot of space tangent X c can't huge x over four plus 3/4 times are integral of C can't cubed, which is tan axe. Seek an axe over too, plus 1/2 Ellen of Seek an Axe plus Tangent X Plus C, of course. And if we do all the math and simplify our fractions here, we get the following plus 3/4 times. 1/2 is 3/8 and that 3/8 is also attached to our log term. And we have our integration constant so again only had to do was applied. The sea can't formula over and over again.

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