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If $ \displaystyle \int^9_0 f(x) \, dx = 37 $ and $ \displaystyle \int^9_0 g(x) \, dx = 16 $, find $$ \int^9_0 \bigl[ 2f(x) + 3g(x) \bigr] \, dx $$

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01:01

Frank Lin

00:27

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Missouri State University

Baylor University

University of Nottingham

Idaho State University

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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04:31

given that we know the integral from 0 to 9 is 37 0 to 9 of G of X is 16 then we want to evaluate this whole expression down here. Basically, you can always add to integral separately. So let's put a plus between these and then the other thing is we want to go ahead and multiply this by a factor of two, which we could just multiply out front, the integral of ffx and same thing with the integral on the right side. Here, let me just move this over and let's go ahead and put a three as a factor there, since that is multiplied by three in front of the integral of G of X. So basically two times all of this is the same thing as just two times 37. So to evaluate this, it's just going to be two times 37 waas three times. All of this is the same thing as three times 16. And when we simplify these two down that wind up giving us a total value of 122. Therefore, the Senate girl here combination of both with the extra factors there evaluates to 122

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