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Evaluate the given integral and check your answer.$$\int \sqrt{s^{3}} d s$$

$$\frac{2}{5} s^{5 / 2}+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 1

Anti differentiation - Integration

Integrals

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

01:44

Evaluate the integrals.

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Evaluate the given indefin…

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Evaluate the given integra…

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Evaluate the integral.…

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Evaluate the indicated int…

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Evaluate the definite inte…

3/Evaluate the integral. <…

Okay. As you look at this problem, we have to do the integral of the square root of s cute D s. So all of these s s not five. Well, I'm gonna have a five. My answer. Eso I need to be clear. What I have my students do is rewrite the square root as to the half power. Um And so the index is what goes into denial American. Then the exponents three goes into the numerator This way Your rule for the anti derivative where you add one to the exponents is easier to figure out with rational, worse aan den, you need to multiply by their super of that new experiments. So we're already ready to find the anti derivative the interval where you add 123 halves. And that's where you get five in your answer So you could see my fives and my s is look very similar. Eso in the numerator is five hash on. Did you have to multiply by the reciprocal? So down here is the divided by five and right there is a s. They're like torturing me with putting an s in the problem and then plus c eso You can check your work by taking the derivative of that E. D s of two of this s to the five past power plus C. And what you'll see is when you bring that explanation front all of those twos cancel the fives, cancel you subtract one from your exponents back to that three house power and the derivative of a constant zero. Which is why we need to have a constant for the anti derivative because it could be anything we don't know what it is. Eso That's why circled in green is your correct answer.

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