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Evaluate the integral. $ \displaystyle \int_0^…

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Problem 29 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int \tan^3 x \sec^6 x dx $


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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Related Topics

Integration Techniques

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Video Thumbnail

01:53

Integration Techniques - Intro

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

Video Thumbnail

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

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Watch More Solved Questions in Chapter 7

Problem 1
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Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
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Problem 45
Problem 46
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Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
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Problem 58
Problem 59
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Problem 68
Problem 69
Problem 70

Video Transcript

this problem is from Chapter seven section to problem number twenty nine in the book Calculus Early Transcendental lt's a condition by James Door. We have an indefinite integral of tangent Cube time seeking to the sixth power Since we haven't even power on the sea can't let's proceed by pulling out, seek and squared We have tangent cute times he can to the fourth which weaken Raya's c can squared square So this is our Sikh into the fourth right here and we still have our remaining seek and squared at the end The reason for pulling out the sea can't square so that if we do it you sub you equal standing Then we see that do you will be seeking square eggs d x So if we want to use the u sub, we'Ll also have to deal with the Sikh hands squared and the parentheses And the way to deal with this and the parentheses is to apply the path Agron identity seek and squared is tan squared plus one. So let's use this identity. Next we have tank you and then in the parentheses applying this But there is an identity in the right. We have tan squared X plus one in the Prentice's That's all being squared. Time sequence Word. Now, at this point, we see that we can apply the U substitution u equals tangent of X. So if we do this so tan Cube becomes you Cube And in the parentheses we have you square plus one that's all squared in the second square x t x That's just do you. So let's go ahead and simplify this Interbrand whether you cued and then in the parentheses, we have you forth plus two, you square plus one and let's go ahead and distribute issue to the third power through the apprentices so that we get you to the seventh to you to the fifth. Plus you, kun, do you and we could evaluate these three intervals using the power rule. We have you to the eight over eight to you to the six over six plus you know, the fourth over four and plus our constant C. And finally we can write our final answer in terms of X by going back started you sub So let's replace you with Tangent of X and we have you to the eighth becomes tangent to the power of X. And we could simplify this fraction too. Over sixes, one over three. So we have tangents to the sixth Power Vicks Laboratory, and then we have you to the fourth over four. So that becomes attention to the fourth all over for you and plus their constancy, and that's our answer.

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Calculus: Early Transcendentals

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Related Topics

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Top Calculus 2 / BC Educators
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Heather Zimmers

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University of Michigan - Ann Arbor

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University of Nottingham

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Lectures

Video Thumbnail

01:53

Integration Techniques - Intro

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

Video Thumbnail

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

Join Course
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Evaluate the integral. $$\int \tan ^{3} x \sec ^{6} x d x$$

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Evaluate the integral. $$ \int \tan ^{4} x \sec ^{6} x d x $$

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Evaluate the integral. $\int \tan ^{4} x \sec ^{6} x d x$

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Evaluate the integral. $$ \int \tan ^{4} x \sec ^{6} x d x $$

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Evaluate the integral. $ \displaystyle \int \tan x \sec^3 x dx $

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Evaluate the integral. $ \displaystyle \int \tan^3 x \sec x dx $

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Evaluate the integral. $$ \int \tan ^{6} x d x $$

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Evaluate the integral. $$ \int \tan ^{3} x \sec ^{3} x d x $$

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Find the indefinite integral. $$\int \tan ^{6} 3 x d x$$

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Evaluate the integral. integral sec 6x tan x dx
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