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Evaluate the given integral and check your answer.$$\int\left(5 x^{2}+2 \sqrt{x}-3\right) d x$$

$$\frac{r^{4}}{2}+\frac{3}{r}+4 r+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 1

Anti differentiation - Integration

Integrals

Missouri State University

Baylor University

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.

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Evaluate the given integra…

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evaluate integrals

So we're looking at the integral of five X squared plus to square events minus three DX. And what I always have my students do is write this as rational exponents. So what that means is changing the square root as X to the one half power. And that's a fact in algebra that all students should know. So now, as we look at this problem, we can follow the rules pretty straightforward, where you add one to your exponents and then you divide by a new exponents. So this constant is still here that has to be divided by your new explosive five divided by three. Um, if you struggle with adding one toe one half, you know, you might off to the side think about how one is the same thing as two halves, so that makes it three halves. And instead of dividing here, what I would do is multiply by the reciprocal of that answer. Um, the anti derivative of three is three x and then you have to have a plus c eso just thio clean this up just a little bit. You don't have to, but to me, um, you know it's worthwhile to rewrite. This is two times two. So it's four thirds X to the three has power minus three X plus C. And so the whole premise of checking the work is, if you did the derivative of this, you will get back to the original problem. And you would because three times five third, just five Subtract one from your explain it. You get to three has times four thirds is to subtract one from your expo and you get one half the derivative of negative three X is negative three and the derivative of a constant zero. So that is how you would check your work. Check. Mark, check, check, check area.

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