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

$$\frac{2}{3} t^{3 / 2}+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 1

Anti differentiation - Integration

Integrals

Campbell University

Harvey Mudd College

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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s O. This is the first problem where we have a t in the problem. Um, and that shouldn't be a big deal, because this is telling you what the independent variable is, so it's almost like everything about in terms of time. Um, so the first thing you want to do is just rewrite this with rational exponents in calculus. We expect you to know that t to the one half hours. The same thing is the square root of team on Ben. It's just a two step process. You add one to your exponents and then multiply by the reciprocal of that new exponents. So when we look at this derivative, I would expect my students to know that adding 1 to 1 half is three halves. But then you have to multiply by the reciprocal of that three have so flip. That's two thirds, and you also need a plus. See, why do you need a plus c? So this is your correct answer. Let me explain, because if you do the derivative of this two thirds t to the three has that should be d d t. Sorry about that. Uh, to the three has power plus C. Well, if you bring this three halves in front, multiply it. You know the two thirds times three halves and you subtract one from that exponents thes twos, cancer. These threes canceled way have t to the one half power. And the director of a constant is always zero eso if we look at the original problem, T to the one hour equals the square root of tea and we don't We don't know if there is a constant for that anti derivative because there's multiple functions that could have a could get us to this. That's why we need a plus C. Um, adding zero is not gonna change the value of it. So circled in green is your correct answer.

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