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Evaluate the integral.
$ \displaystyle \int \sin \sqrt{at}\ dt $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 5
Strategy for Integration
Integration Techniques
Oregon State University
Harvey Mudd College
University of Nottingham
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.
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Let's start off here by doing you use up that here, Do you? Using the chain rule from capitals. We could rewrite this as a over to you DT. And let's go ahead and solve this equation for DT. So we have dt equals to you over a to you and that's whole replace over here in the original integral sign you replaces the first term and then here for DT We can use this so we have a new integral and let's pull out that constant. So on the downside, we do have this extra factor in front of the sign that we didn't have in the original. But the expression inside the sign is now much easier to the OS. So for this in a girl here, you can do integration. My parts Let's go ahead and take w to be you t w equals Do you t v equals sign you do you the equals negative co sign you and then you have to over a recall the formula for integration by parts You've even this case w v minus inner growth e d w. So using that formula here, there's a W times me and then in a girl Z and then DW So this is just negative to you, coz I knew over, eh? Plus two over, eh? Sign you And that was good. And add that sea and finally and the last step here is suggest go to your original substitution and took plug in okay with to plug in U equals radical eighty season. So that's the first term and then for the second term, and that's your final answer.
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