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Evaluate the integral by substitution$$\int x \sqrt{x+1} d x$$

$$\frac{2}{15}(x+1)^{3 / 2}(3 x-2)+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 9

Two Integration Techniques

Integrals

Baylor University

Idaho State University

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.

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here. I want to integrate X times the root of X plus one. And to do that I'm going to u substitution and I will let you equal The X-plus one in the square root sign and I'll work out to you by the X. She's one which tells me at once that D U and D X are the same. So I then replace everything I see here with new terms. So the X Will be U -1 from the first line. So I have integral of u minus one square root of you and dx and the you are the same in this question. So we have here then if I just expand the brackets out U power 3/2 minus you? Power one half do you. And that's a very simple integration. So I get you 5/2 times 2/5 minus you. Power 3/2 times two thirds plus C. Now I make this a bit simpler algebra. I would take out to 15th you 3/2 as a factor and see what's left. We'll be three here to get the 2/5 and just you To get the power of 5/2 minus Only five here to get 2/3 and I'll be it plus C. The last step is to replace the U by its value of X plus one. So it's to 15th x plus one. Our 3/2. And here we have three times X plus one minus five plus C. And the answer will be to 15th X plus one. Power 3/2 Times three x plus 3 -5 -2 a c, and that he is the answer.

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