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Evaluate the integral.$$\int \tan ^{4} x \sec x d x$$

$\frac{1}{4} \sec ^{3} x \tan x-\frac{5}{8} \sec x \tan x+\frac{3}{8} \ln |\sec x+\tan x|+C$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

PRINCIPLES OF INTEGRAL EVALUATION

Section 3

Integrating Trigonometric Functions

Integrals

Integration

Integration Techniques

Trig Integrals

Trig Substitution

Missouri State University

Harvey Mudd College

University of Nottingham

Boston College

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Hi. Some of this question we're going to be solving for the tan of X to the fourth times the seeking of X, t X and so from here, what we're going to do is we're just manipulate our equation a little bit and we're going to have the tan of X squared times he tan of X squared times the Sikh end of acts d x on from here. We're gonna use the trick. Equations were going to use the trick equation that states that the tan of X squared plus one is equal to the Seacon of X squared. Andi So the tan of X squared is equal to the seeking of X squared minus one. Now we're going to just right this into our equation. Substituting in are seeking of X squared minus one for all the chance in the equation. So we're going to now have the seek and of X squared minus one time, See, seek in of X squared minus one times the seek it of X d x. Uh, now from here, we're gonna just distribute, so we're gonna have the integral of the Sikh in of X to the fourth minus two times the Sikh int of X squared plus one times the seek and of acts X on from here. We're just going to, um, distributed through our sequence would have seek IND of acts to the fifth minus two times the Sikh int of, um X cubed plus the seeking of x d. X on from here. We're gonna split this into three different integral. So we're going to have the integral of the stick in of activated three F's minus the axe. Sorry, Linus, to times the integral of the Sikh and of exe cute d X plus the integral of the Sikh and of X de X. And now from here, we're gonna be, ah, using a couple of equations. So, um, first thing we can do is we can solve for, um, are seeking of X to the fifth. Using the equation that states that the integral of the second toothy on of Axe D X is equal to the speak int of a new minus to x 10 x over and minus one plus and minus two over on minus one times the integral of the Sikh int of X of on my Penis to D X and so we're going to have her our first integral. We're gonna have be, um, seeking of and minus two. So five minus two, which is three. So the secret of X cute times the tan of acts over five minus one. So four plus five minus two, which is three over five minutes for one which is four times the integral of the Sikh int of X to the third or the seeking of X cubed d X Andi. Now we're going to used the There's an equation actually, for this Seacon of x cubed and we're gonna be using that. Just offer next one So that one states that the secret of X cubed D X is equal to one half times the seeking of acts Tan X plus one half times the Ellen plus the tan of acts. Um, on then from here, we're going to be using the another equation for the seeking of X dx. And that equation is just going to be plus the I'll on of the Sikh int of X plus thicket tan of x. And of course, we have a plus C For all of these, the each would have an individual C but it would all just end up being one. See anyway, um, so I'm just gonna one see there so we're going to multiply by each of these by two. Negative to actually said, this is going to cross out and so we could make this and negative and we the negative there. So that is going to cause these two to be able to cross out. And so we're going to be left with e seek IND of X cubed and fax over four plus 3/4 of the integral of the Sikh in of X cubed D X minus the Sikh end of X tan of X plus. See, And so from here, or going to do is we're gonna use the same equation that we used to solve for the secret of three x dx Teoh Saul for are integral. So now we're going to have the second of X cubed times the tan of X over four plus 3/4 of one half the second of axe of acts, plus one half of the Ellen of the second of acts, plus each hand of X. And then this is going to be minus be seek and of X, take a fax plus C. And so from here we can just simplify. So we're going to have the seeking of acts cubed times the tan of acts over fourth plus 3/8 times the seek and of X Tanna Axe closed 3/8 times the on of the seeking of X plus the tan of X minus the seek and of X tan of X plus c. And now we can make this instead be eight over eight. And so we're going to have as our final answer the second of X cute times the tan of acts over four, my Penis five AIDS of the Sikh and of x times, the tan of X us 3/8 of the L on of the Sikh and of acts plus the tan of Axe plus see And that is going to be our final answer

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