Problem

If $ \displaystyle \int_0^{\frac{\pi}{4}} \tan^6 …

05:58

Problem 49 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int x \tan^2 x dx $

Answer

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

Topics

Integration Techniques

Book Cover for Calculus: Early Tra…

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Discussion

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Video Transcript

here we have a mineral of X times tangent squared of X. Let's go ahead and use integration. My parts Let's take unit B X so that do you is the ex. Then we're left over with d V equals Tan Squared X, which we can also write using our pathology and identities seek and squared minus one, and doing writing it this way makes it easier to find the So it's a fine B which is integrate so the integral C can't square is tangent and the integral of one is X. So here, using parts U V minus integral medio we have ext time San Genetics minus X squared. So this is you need minus inaugural video. So there's our you ve and over here underlined is in a girl video. All right, for the next step, let's just rewrite this u b we have x ten x minus, X squared minus and our world So it's quite and bright. This is two separate minerals and then here we have a double negative. So have a plus into a Rolex I couldn't separate. Some are scratch work. So now we know how to evaluate these inner girls. Xed annex minus X square so anti derivative of tangent is natural law of seeking absolute value of seeking. You could either attain this result from your table or here you can go ahead and use it. Use up. So first you can right tangent equals sign of our coastline. And then you could do it, you substitution. That's another way. Toe. Evaluate this anti derivative and dance I derivative of X. We're exclude over, too, and let's add our constancy. So the last thing we can do here is combined like terms. We have a negative X square plus X squared over two sorts commandos. So doing so we have negative X squared over two plus X tan X minus Ellen. Absolute value seeking a Vicks plus he and there's our answer.

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J H.

Topics

Integration Techniques

Book Cover for Calculus: Early Tra…

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Top Calculus 2 / BC Educators

Catherine R.

Missouri State University

Anna Marie V.

Campbell University

Kristen K.

University of Michigan - Ann Arbor

Samuel H.

University of Nottingham