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Evaluate the integral.

$ \displaystyle \int \sqrt{\cos \theta} \sin^3 \theta d \theta $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

Christopher G.

September 11, 2022

The math is done incorrectly when the instructor "'attempts" to distribute u½ across (1-u²) which he does incorrectly and so the entire solution to this integral is incorrect.

Missouri State University

Campbell University

University of Michigan - Ann Arbor

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.

03:17

Evaluate the integrals.

03:10

Evaluate the integral.…

00:46

Evaluate the indefinite in…

07:21

05:51

02:40

09:50

This problem is from Chapter seven section to problem number thirteen in the book Calculus Early. Transcendental. Lt's a condition by James Store. Here we have the integral of the squared of co sign of data times San Cute of data. So first, let's rewrite this co sign this crude. Of course. Sinus cosign data to the one had power. Then for the sign. Cute. Let's re write this sine squared data time signed, Ada. Then we could use a pathetic an identity to rewrite sine squared as one minus co sign squared. So we have an integral coasts and data to the one half Power one minus co sign squared data times signed data and now we see we're ready to apply u substitution. So here we should take you'd be co sign of data. So that to you is negative scientist the equivalently negative. Do you? It's scientific data. So after applying to see substitution, we can rewrite this integral as thie integral of you to the one half power. Yeah, one minus you square, do you and we need a negative to you. So let's pull this negative outside of the rule. So now let's distribute issue to the one half power inside the apprentices. Whether you two, the three half power my tissue to the five over to power. And now we could evaluate those inner girls. And we have you threw the five paths times two over five, minus you seven halves types to over seven. And let's at the constant C of integration. And then here the last step is to replace you with co sign a potato. And we could also distribute the negative signs as well. So we have a negative two over five cosign data to the five half power, plus to over seven cosign data to the seven half power plus see, and that's your final answer.

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