Problem

Show that for any integer $n>1$, we have the redu…

08:43

Problem 32 Hard Difficulty

Evaluate the integral using both substitutions $u=\tan x$ and $u=\sec x$ and compare the results.
$$\int \tan ^{3} x \sec ^{4} x d x$$

Answer

$=\frac{(\sec x)^{6}}{6}-\frac{(\sec x)^{4}}{4}+c$

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Calculus 1 / AB

Calculus 2 / BC

Calculus: Early Transcendental Functions

Chapter 6

Integration Techniques

Section 3

Trigonometric Techniques of Integration

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Video Transcript

So we want to integrate tangent cubed X times seeking to the fourth X with respect to X using two different methods the first time through we want to use U substitution with you equaling tangent of X And before we let you equal tension events let's go ahead and rewrite our an original interval as Tanja Cube decks times seeking square next time seeking square necks. So what we're doing there is word extrapolating seeking squared X from the secret to 1/4 X or another way to think about that is we just factor seeking to the fourth of X into two squares. So if U s tangent of ETS, then bu the derivative attention Ex is seeking square Novaks TX So going back to substitute in we have a substitution for Tangin X to be identical to you. So I have a YouTube that was seeking square decks that will come back and deal with in a moment and then the Seacon Square Next e x becomes our dean. You we still don't have an integral completely in terms of you, so we're going to real right Seacon Square decks using the Trigana metric identity that one plus tangent squared of X Equal Seacon Square to vex some rewriting Secret square Novaks as one plus Tangent Square to vets, we now can rewrite the tangents as a year. So now we have the interval of you cubed times the quantity in one plus you square with respect to you must distribute obtained you cubed plus you to the fifth. But there expect to you next We will use the power rule for integration well at one to the power and divide brother New power. So we have one for a few to the fourth plus 1/6 year to the six That's a constant back substituting again since we had that you was tangent of X We now have went forth times tangent to the fourth of X +16 times tangent six of x, plus a constant. Next let's compute the same integral letting you equal seeking x. So if you was seeking next then we know they BU is seeking x Tangin X he acts, which means that we need to rewrite our original interval so that we can replace some of the other into t. So we know that tangent to Dex is Tanja Next times Tangent Square. Next. So we need a tangent x as part of the substitution for do you and will rewrite the tangent Square decks in terms of seeking X momentarily and then seek into the fourth Ex is seeking next time seeking cute decks again We need another seek and x to go along with the tangent X for D So we're gonna extract late to see connects from the secret to a four to get a secret next time seeking cute x dx So rewriting we technically have seeking Next time standing Next times TX which I'll write together The tangent squared x since we have the following identity one plus tangent squared X Seacon Square decks because subtract one on each side to see that tangent Square decks is equivalent to seek in squared X minus one. So rewriting tangents where decks in terms of sequins have Seacon square necks minus one for tangent square decks. We also have the seeking que decks. Now it's in the form weaken Do direct substitution Seeking X was equal to you. So now we have a use squared minus one. Take that 1 to 10 times you cube to send see connects was equal to you, and we haven't expression for seeking extension. X DX is now 18 years. Let's distribute, Try octane you to the fifth minus you cute with respect to you integrating term by term, adding one to the power and divided by the new power, we have 16 year to the six minus 1/4 you to the fore. And let's use K in this case for her constant. But we're not done just yet. We know the back substitute, recalling that you was seeking XO have won six times seek it toe the six of X minus one for seeking to the fourth of X plus r Constant K. Next, let's show that the two answers are equivalent, so I'm going to use our second answer and show that it's equivalent to our first answer. So using 16 seeking to the six sets minus 1/4 seeking to the fourth of X plus her constant K, let's recognize that seek into the six X is equivalent to having seek it squared of X quantity. Huge because when we take up our race or power, we multiply the powers. Showtime's threes of the point of power. Six. Similarly, for seeking to the fourth of X, that would be equivalent to seek in squared of X. When it is clear, the reason we want to look at the sequence squares is because of our identity. For Seacon Square being one plus tangent Squared X and since our first answer had tangents, we want to convert our second answering to tangents so recognizing that Seacon Square necks is one plus Tangent Square debts. So now we need a cute that X pressure minus 1/4 times, according to T one plus Tangent Square necks. It's cleared closed for constant K, so we can use some pre calculus techniques to Cuba one plus tangent square decks. So we have 16 times one cubed plus one square times tangent square to vex, plus wine times, tangent squared of X cleared plus tangent squared of X cute and that will have additional Constance by our Pascal's triangle to give us an additional constant of the three for the second term in a three for the third term. So we're using Pascal's triangle with the expansion so we have the constants. 133 and one. It was insert extra constant of three. It's a constant of one of three. Again sorry and then next we have minus 1/4 can use the same expansion technique. One square is one plus two years in Pascal's triangle. Again, it would be the extra constant times tangents squared elex and then finally plus tangent to the fourth of X and then plus R. K. On the end. Cleaning that up, we now have 16 plus 1/2 times stain that squared of X plus one halftime standard to the fourth of X +16 times Tangent to the sixth of rex, minus 1/4 minus 1/2 tangent Square Durex minus 1/4 Tangent Tow fourth of X. That's our constant K. Next. Next, recognize that we have a 16 minus a 14 plus a constant K. Let's take all those constants and add them together and just call them. See. Use the same coloring so you can see the relationship there so those three constants together will give us a seat and then let's combine like terms for the others. So we have a positive 1/2 tangent, square necks and a negative 1/2 tension square decks so they add up to zero. Next would have a 1/2 tangent to the fourth X and a minus 1/4 attention to the fourth Ex. But we think of 1/2 is being a to force. We have to force change until four minus 1/4 Cancer ho, for which will give us a 1/4 tangent to the fourth ex and then we also have Plus are +16 changes to the six X, which all right first, since it's the larger power so have won six tangent of the six X +14 of attention to the fourth, That's plus R. C. And we can recognize that that's the same answer that we obtained from the first. Nothing went forth attention to the fourth That's +16 tantra toe six x plus C

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