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Evaluate the integral $\int \sqrt[n]{a x+b} d x,$ and then use integration by parts to evaluate the integrals.Exercise 5

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 9

Two Integration Techniques

Integrals

University of Michigan - Ann Arbor

University of Nottingham

Boston College

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.

04:47

Use integration by parts t…

02:20

Evaluate the integrals.

02:03

Evaluate the integral.…

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Evaluate the given integra…

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Use a substitution to eval…

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evaluate

so we want to find the following integral. Using integration by parts. So I've got the formula over here to put right. Um So in our formula we need to choose a you and a D. V. That makes up this entire and a role here. Um So let's let you be equal to X. And let's let DV Be the rest. So we can rewrite this as two X -5 to the negative one half power. Mm hmm the X. So now we need to differentiate you. Um So d'you is just going to be equal to dx and we need to integrate both sides over here to get ve so V is going to be equal to the integral uh two X minus five to the negative one half power D. X. So to solve this integral we need to use some sort of U substitution. But we already have you going on here. So let's uh use a different variable. So let's let z be equal to two X -5. Then dizzy is equal to two D. X. So now we can rewrite are integral. We've got the integral of the sea to the negative 1/2 power. And then in order to put a DZ here we need to put it to so we can multiply that out front. Um So we've got to times the in the role of Z to the negative one half. So we add one to the exponents Z. To the one half. Um And we need to multiply another two out front there. So this is equal to four square root of Z. So now we have everything we need we can use our formula. So our original integral here is equal to U. V. So X. Times for square root dizzy. Uh Sorry we don't want to keep seeing here. So yes this is for scrubber dizzy but we uh need to substitute X. Um back in in terms of X. So this is four square root of two X -5. So let's write that here two X -5. Uh that U. V minus. And then we need to integrate V. D. You so V. This four can come out front and then we've got square root of two x -5. Mhm. Uh D. U. Which is D. X. So now we just need to integrate this guy, let's rewrite our first term four X. Square root two x -5 minus up to the side here let's do our integral Square root of two x -5 D. X. Um So we need to do another substitution. We can use anything. So let's Collins Odetta Be equal to two X -5. So again D Data is equal to two d. x. Um So are integral becomes the integral of square root of data. And to put a. D. Data here, we need to add this to out front. So this is going to be equal to two times in a role of data to the one half. So what do we have enough data to the one half plus one, which is three halves. Then we need to multiply by 2/3 out front. So this is going to be equal to 4/3. We can replace data with our original two X -5 Uh to the 3/2 power. So that's going to be our inner girl here. So we've got four Times This 4/3. So we really have uh 16 3rd times two X -5 to the 3/2 power. And then we have to add our constancy.

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