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Evaluate the integral.
$ \displaystyle \int_0^{\frac{\pi}{2}} \cos 5t \cos 10t dt $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 2
Trigonometric Integrals
Integration Techniques
Missouri State University
Campbell University
Oregon State University
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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this problem is from Chapter seven section to problem number forty three in the book Calculus Early Transcendental Sze eighth Edition by James Door We have a definite in a rule of co signed five tee times Cose I intent e here it will be useful to use a formula for the product of two co science. In our case, we have a formula from the text and in our problem, the value of a five t and we could take b equals tensy. So using this formula pulling out the one half we have one half Indian girl zero power too coastline of a minus B So it's a five t minus twenty plus coastline of a plus B. So it's five. Teen plus twenty. Okay, so observing the first co sign here we have a minus five t so we can replace co sign of minus five t with co sign a five t since coastline is he even function not necessary to do the step But it will simplify things a little bit for us. So we have one half and a girl zero power too cosign five t plus coastline fifteen t and my help you to use the U substitution here to evaluate these inaugurals. You could take you to the five T and here you can take you two be fifteen t. Yeah. Evaluating these generals, we have one half signed by T over five. Plus I kn fifteen t over fifteen and points are zero in power for too. Let's go ahead and plug those in for tea. So we have one half sign of five pie over, too. On princesses, plus sign of fifteen power, too. That's also in Prentice's over fifteen. So this is from plugging in Pi over till for team. And when we plug in zero for tea, we have sign of zero over five, plus sign zero over fifteen. And we know from the unit circle tables of these terms will be zero no, also from the unit circle. We know that sign of five pilots who is one. So we have one half. We have won over five and then sign of fifteen pirates who was minus one. So we have a negative one over fifteen, and this can be simplified to give us two over thirty, which is one over fifteen. And that's our answer
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