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Evaluate the integral.

$ \displaystyle \int_1^4 \frac{e^{\sqrt{t}}}{\sqrt{t}}\ dt $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

Integration Techniques

Missouri State University

Campbell University

University of Michigan - Ann Arbor

Boston College

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

01:52

Evaluate the integral.…

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Evaluate the given definit…

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Evaluate integral

Let's use a U substitution for this animal. Take you to be square. Rarity then do you one over two times a radical T or licious? Multiply that two over and we get DT over square of T. And that's precisely what we see over here in the original problem. So let's go ahead and rewrite this integral in terms of our new variable. So let's watch out for those limits of integration, we have a one there, so t equals one so plunged it into this equation. T equals one, so you equals one. So our lower limit will still be one. Now, check that upper limit too. So here t equals four. So you was too. So this is our new upper limit and our new variable now and then we just have e to the u Do you either The U accounts for this term up here and then, as we mentioned, do you and we forgot Forgot the too. So let me put that to our here so that this to do, you will account for the remaining part dt over the radical and and now just generate this. So that's just to eat the you wanted to. So to ease where minus to eat. And then there's you can factor a few if you must take out to you a minus one, and that's a final answer.

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