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Evaluate the integral.

$ \displaystyle \int \tan^2 x \cos^3 x dx $

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$\frac{\sin ^{3} x}{3}+C$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

Oregon State University

Harvey Mudd College

Idaho State University

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.

02:41

Evaluate the integral.…

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07:21

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this problem is from chapter seven section to problem number sixteen in the book Calculus Early transcendence, ALS a condition by James store. And here we have an indefinite integral of tangents where times co sank you. So first thing we can do is use the definition of tangent tangent assigned over co sign so we can rewrite Tan Square is sine squared X overcoats and square Becks. And here we see that we could cancel off some of these co sign so we could cross off these two in the denominator, we could take off two of the coastlines in the numerator, and then we're just left with Cho since the first power. So that gives us in a girl sine squared X times close antibiotics. So here we can use the u substitution take you to be signing books so that do you is co sign of X d X. And then our general simply becomes the integral of use. Claire, do you? Which we can evaluate using the power rule. We had you cubed over three plus e. And then for the final answer, we should go back to our u sub in back substitute from you back into sign of X. And there's a garment. Final answer

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