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Evaluate the integral.

$ \displaystyle \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{x}{1 + \cos^2 x}\ dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

Integration Techniques

Missouri State University

Campbell University

Harvey Mudd College

Boston 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.

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Evaluate the integral.…

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Evaluate the indefinite in…

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Evaluate the definite inte…

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So here let's define the int'l grant Bye after Lex. And let's note that f of negative X equals here we have negative X one plus co signs where negative X Now we could pull out this minus outside and then using the fact that co sign is even so here we were using the fact that Coulson is even we end up with negative FX. So this shows us that efforts are f is an odd function. So graphically, this means that the graph is is symmetric about the origin. So get this is not the graph of our function. But it does have this property that it symmetric about the origin. And then because if you look at her in points negative, Piper soon a pirate or two. So due to the cemetery around the origin, the a positive area that you get on one side cancels out with the negative area that's on the other. And this is coming from a theory, um, that we've seen earlier in the textbook that says that if you have a integral of this form, if if his are since we showed, f isn't our function, this in a girl has to be zero. And that's our final answer.

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