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Evaluate the integral.

$ \displaystyle \int \tan^4 x \sec^6 x dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

Missouri State University

Campbell University

Oregon State University

Harvey Mudd College

Lectures

01:53

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.

27:53

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

07:16

Evaluate the integral.…

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Find the indefinite integr…

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01:20

this problem is from Chapter seven section to problem number twenty five in the book Calculus Early. Transcendental lt's a Tradition by James Door Here we have an indefinite a roll of tangent to the fourth power of EXT. Time sequence of the Six Power of X. So the first thing we can do here is rewrite the sea cans of the six power. So let's write as seeking Square X square. So that c cancer, The Fourth times seeking Square of X, the ex. And now we can use one of our protagonist identities to rewrite this Sikh and squared of X. And we have integral ten to the fourth power, vex parentheses and then seek and squared. Using our battalion identity becomes tan squared X plus one and that's also being squared time seeking and square eggs. The ex. Now we see we can apply a u substitution. Let's take you to be Tana Becks. So then do you become sick and square? Vicks the ex? So here we see. We have to you ten to the fourth is simply you do the fourth and then we have use flair plus one in the parentheses. And that's also being square. So our inner girl becomes after applying the substitution you to the fourth power you squared plus one square, do you Before we integrate, let's go ahead and simplify. This is much as we can. We have you to the fourth. So let's evaluate this square. We have you to the fourth and the parentheses. Plus two, you squared plus one and let's go ahead and distribute issue to the fourth through the apprentices. So have you ate Plus, to you to the six Clinton plus you know, the forth to you. And now we can apply the power rules. Evaluate each of these three intervals. So doing So we have you threw the knife power over nine plus to you to the seven number seven. Plus, you know, the fifth power over five plus our constant of immigration. See, And finally we come back to our original U substitution so that we can back substitute you with Tanox so that we have ten to the ninth Power Becks Overnight plus two, ten, seven. Power affects over seven, plus stance of the fifth power of X over five and plus our constancy. And there's our answer

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