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Calculate the average value of $ f(x) = x \sec^2 x $ on the interval $ [0, \frac{\pi}{4}] $.

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 1

Integration by Parts

Integration Techniques

Missouri State University

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.

02:04

Calculate the average valu…

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04:29

Find the average value of …

00:48

04:09

Calculate the average over…

01:51

00:58

00:52

The problem is calculated, the average value of f of x is equal to x times, second x squared and the interval from 0 to pi over 4 n. We reach value of f of x and this interval is equal to integral from the oro to pi. Over 4 x, second x, squared dx over pi over 4 point, but this is equal to 4 over pi times integral from 0 to pi over 4 x times. Second x, squared dx, but this integral we can use integration by paste formula is integral from a to b. U, o d is equal to u v from a to b minus the integral from a to b upon for or for our problem we can write. U is equal to x and prime is equal to second x square. Then up is equal to 1 and v is equal to tandant x, that this is equal to 4 over pi times. U times so. This is x times 10 and x from 0 to pi. Over 4 minus the integral up times, le satis is tandant x from 0 to pi over 4. So this is a calculefour over pi times, pi over 4 minus integral from 0 to pi over 4 sine x over cosine x dx. So we can use us of stitution that y is equal to cosine, x and y is equal to negative x x. This is equal to 4 over pi times, pi over 4 minus integral from 1 to root 2 over 2, and here this is plus dy over y. This is equal to 4 over pi times pi over 4 plus n y from 12 to 2 over 2 poias. The answer is 4 over pi times pi over 4 plus n nuative 2 over 2 point. So this is the result.

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