Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Evaluate the given integral and check your answer.$$\int\left(\frac{5}{x}-2 e^{x}+7\right) d x$$

$$5 \ln |x|-2 e^{x}+7 x+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 1

Anti differentiation - Integration

Integrals

Missouri State University

Campbell University

Harvey Mudd College

Baylor University

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.

01:24

Evaluate the given integra…

02:22

02:02

Evaluate the integral.…

01:35

03:39

Use a substitution to eval…

02:03

Evaluate the integral and …

01:15

Evaluate the integral.

03:21

Find the integrals .Check …

All right, so we're finding the integral of this. Um, this function five over X minus two e to the X. Okay. Plus seven DX throwing some parentheses there for you. And, uh, the main theme in this. So just off to this, I'm gonna talk about the theme. Is that because the derivative of natural log of X is equal to one over X, that's gonna be the anti derivative of this. Now, notice that if I had a constant in here like see, so in this case, five b c over X. So we're working backwards on that one, and in the same manner, the derivative of I'll use K for a different constant E to the X is equal to K E to the X. So when I'm ready to do this and I derivative just from the get go, um, that the anti derivative of five over X would be five times natural log. Now, you do want to make sure you do absolute value of X because we can Onley log positive numbers and it's not clear if X is a negative. So we want to make sure we have absolute value of that on then the derivative A B to the Izzy to the extra going backwards is gonna be the same thing. So it's still minus two e to the X. The anti derivative of seven is seven X, and that should make sense because the derivative of seven X is just seven. But then you also have to have a plus C, and that's what they mean by check. Your work is take the derivative of your correct answer, and what should happen is you get back to the question that they asked you and we do. That's how you check your work.

View More Answers From This Book

Find Another Textbook

02:04

Evaluate the given integral.$$\int \frac{e^{2 x}}{\left(e^{2 x}+1\right)…

01:32

Evaluate the given integral.$$\int \frac{e^{2 x}}{e^{2 x}+1} d x$$

02:08

Sketch the area represented by the given definite integral.$$\int_{-1}^{…

03:45

Determine the area of the indicated region.Region bounded between $f(x)=…

04:25

Find (a) $f_{x x}(x, y),$ (b) $f_{y y}(x, y),$ (c) $f_{x y}(x, y),$ and $f_{…

02:31

Evaluate the given integral.$$\int\left(2 x^{2}-3 x y^{3}+y^{2}+3\right)…

00:45

Determine the region $R$ determined by the given double integral.$$\int_…

04:12

02:21

Using the result of Exercise $31,$ evaluate (a) $\int e^{2 x} d x,$ (b) $\in…

05:15

u(x, y)$ is a utility function. Sketch some of its indifference curves.$…