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Problem 30 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int_0^{\frac{\pi}{4}} \tan^4 t dt $




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Video Transcript

this problem is from Chapter seven section to problem number thirty in the book Calculus Early Transcendental lt's a Tradition by James Store. We have a definite rule from zero power for abstention to the fourth power of tea. Here we can rewrite this tension to the fourth by pulling out of it Tangent squared. So all we did was re right there. Attention to the fourth power. And moreover, we can replace one of these tangents squares using the pathogen identity seek and square is tension square plus one. So that means tangent squared is seeking and squared minus one. So doing so we can rewrite are integral. We have seek and square T and apprentices minus one kitty. And at this step, we could go ahead and distribute its hand square through the apprentices so we have tan square times. He can't square this integral can be evaluated using a use up. So let's maybe split this up into two other girls. So we have this inner girl minus the inaugural from zero. However, for of tangent squared times one which is just tangent squared of tea, TNT. So we'LL have to deal with these inner world separately because we can use a use of for the first interval. Not quite for the second one, at least not right away. So for the first integral it, maybe the notice in a rule in green. Let's do it. Use up here. Let's take for the green and the girl. Let's take you to be Tanox, our TNT. So that deal was seeking square T. Dante. And since we're dealing with a definite or girl left, it changed those limits of integration. So the new lower limit the lower limit for the new integral will be tan zero, which is still zero. The upper limit will be ten apart before which is one. So this first interval becomes and a girl from zero one you squared, do you? And for the second integral, we could use the Pythagorean identity once again. So we write this tension square. So let's do that. So a tangent squared using our pathetic and identity again Over here on the right. That's just seek and square T minus one DT and we actually can't evaluate the animal. It's the integral C can't squared is just handed. And general one is tea. So no more you suffer Any other techniques are needed here. So now we're ready to integrate over here and uncle of you squared using the powerful web you cubed over three Skittles and point zero one. And for the second inaugural Deanna Girls he can square is tangent We have tangent lt Minus T and those end points See you in a bar before and now we're ready to go ahead and just plug in the end point. So for the first and the girl, we have one over three minus zero When we plug in the end, points one and zero then we have this minus sign over here So we have a minus in between then we'LL plug in pi over four It's a tangent of pirate for is one minus t which becomes power for and now we'LL plug in the end point zero tension of zero zero and we have minus zero. So there we just have subtract zero. So we have one over three minus one minus power before what? Which we can simplify his power before minus two or three. And there's our answer

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