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Evaluate the integral.
$ \displaystyle \int \frac{1 + \sin x}{1 + \cos x}\ dx $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 5
Strategy for Integration
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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Evaluate the indefinite in…
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$$\text { Evaluate } \int …
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Evaluate the integral.…
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for this integral, luscious Tara. I re ready and then multiplying the numerator and denominator both. Bye, Let's do one minus co site. What So schools and simplify that. So multiply out numerator and the nominees are separately on the bottom one. Minus co sign squared Attack one minus cosa. Next, Plus I kn minus sign times Co sign running out of room here times Cosette Nix. And here we can rewrite. This is Science Square. That's one of your dragon identities. So it's good and split this into foreigner rules. What? So the first Herman over sine squared, then co sign science weird on the bottom for all these terms, then here we have sine X over science. Where do you see that? You could cancel one of those signs and then finally, our last term side Times co sign science Weird again. Cancel one of those signs here is Well, oh, so let's go to the next page and then simplify each of these fractions. That's the first thing, and we're off to re writing it. Second integral, you can go ahead and do a use of there if you want to, or you could just rewrite it as co tangent times because he can. Then the next one. This was the one over sign cause he can't minus. And then we have co sign over, Sign attention. And we know the answer to each of these these air just tricking the rules. So here, for the first one, we know it's negative. Attention X, then hear this week, plus course he can. And then over here we get a minus ln plus coach engine. And then over here we'LL have a minus natural log of sign. Let's add that constancy and and there's a final answer.
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