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Evaluate the given integral.$$\int e^{2 x} \sqrt{1+3 e^{2 x}} d x$$

$$\frac{1}{9}\left(1+3 e^{2 x}\right)^{3 / 2}+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 3

The Substitution Method

Integrals

Missouri State University

Campbell University

Oregon State University

University of Nottingham

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.

01:12

Evaluate the integral.…

10:54

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evaluate Integral

Evaluate the integral.

the derivative of each of the two X is two times each of the two X. So if we take you to be the inside function which is a function side of the square root, then do you will be equal to six times E to the two x dx. And now we can divide both sides by six. To get d you over six is equal to eat the two x dx, which is what we wanted because now we can replace eats the two x dx I do you over six rewriting this integral as 1/6 times the integral of you to the one half which is equal to 1/6 times to over three times you to the three halfs. But you is one plus three times e to the two x all raised to the three halfs plus c. And now we can simplify this. That's right. This as 1/9 times one plus three times each of the two x raised to the three halfs plus c. Yeah, and that completes it

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