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Evaluate the integral.
$ \displaystyle \int \frac{\sec x \cos 2x}{\sin x + \sec x}\ dx $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 5
Strategy for Integration
Integration Techniques
Missouri State University
Campbell University
Harvey Mudd College
University of Nottingham
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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03:08
So here let's just er off by multiplying topping denominator. Bye. Let's do it to co sign so many ways to do these types of inter girls. But sometimes multiplying top and bottom by the same sir can help simplify. And the reason I'm choosing co sign is to cancel out with the sea. Can't hear. So we recall that C can't is one of our coastline. So these terms here will cancel out. And so we see Jin and talked. We get too close I into x No t X still and then in the denominator we just have to sign eggs co signed X plus. And then when we multiply these two together, it's just too because of this fact Over here and just for convenience, let me rewrite this. It's just going to simplify things a little bit for us. Let me write. This is sign to X. I'm just using the double angle formula here and then let you be the denominator plus two. Then do you is using the chain rule. But you can see this to you is exactly the same bizarre numerator up here so we can rewrite this inaugural as one over you and then do you? That's just natural log of absolute value plus e. And we can write. This is natural Log signed two eggs plus two plus he and the reason absolute value is not needed here is because sinus between negative one we want so that what we add two side plus two is bigger than one. So in particular is a positive number, so we don't need absolute value here. And because of that, here's a final answer.
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