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Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. $ \displaystyle \int \frac{\sin 2 \theta}{\sqrt{5 - \sin \theta}}\ d \theta $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 6
Integration Using Tables and Computer Algebra Systems
Integration Techniques
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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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Use the Table of Integrals…
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Okay, so this question wants us to evaluate this integral. So to do this, let's use the identity. That sign of Tooth Ada is to Science Data Co signed data. So this tells us that are integral becomes two times the integral of sine theta co sign data divided by square root of five minus sign data dif ada. So now from there, we can substitute this to make it into a more digestible form. So let's pick you equal to sign data. So that means that D u. Is co signed dated data, which means are integral turns into two times. Well, scientist, it is you co signed a dictator is do you over square root of five minus you. And then this form of an integral is exactly this formula from the back of the book. So we have a equal of five and be equal to minus one. So that means are integral is equal to keeping that factor of two in front to over three times b squared times. Bu Sorry. Yeah. BU minus two a times the square root of a plus B you, which is original expression plus c. So then if we multiply all this out, we get 4/3 times negative. You minus 10 square root of five minus you, plus C and factoring out a negative one from that first term, we're left with this form. So now all we have to do is plug in science data for you, and we're done.
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