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Evaluate the integral.

$ \displaystyle \int \sin^2 x \cos^3 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

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.

01:56

Evaluate the integral.…

06:02

02:00

02:46

01:28

This problem is from Chapter seven section to problem number one from Calculus Early Transcendental Sze eighth Edition by James Door And here we have the integral of sine square times coast and cubed of X t X. So the first thing we can do here is we observed that we have ah, odd power on the co sign. So let's first rewrite this by pulling out one of the factors of co sign. So here we are. Rewrite Cho San Cube as co sign squared times call sign. The next thing we could do is apply it. Pythagorean identity to rewrite this co sign squared is one minus sign square. Okay, so that's coming from. But dragon identity that says sine squared plus co sign squared equals one. And then now, after the stuff, we basically see that the problem is tailor made for you Substitution. So here we can apply the use up u equals sign of X so that do you becomes cosign a Vicks the ex. Okay, so then after the step, we have the integral you square one minus you squared Deal. So here we can distribute the power of you square toe one and minus. You squared before we integrate. So we have you square minus you to the fourth, do you? And now we could proceed to integration. So here we applied the power Rolls Royce. So we have you knew that their power over three minus, you know, the fifth hour over five plus c. So we have evaluated the integral. But now she's back. Substitutes are original variable x. So we simply rewrite you, Cube. But using our U substitution. So this becomes Sang Cube. And similarly, you fit becomes signs of the fifth plus e And that's where in the rule

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