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Evaluate the integral.

$ \displaystyle \int \csc x dx $

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$-\ln |\csc x+\cot x|+C$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

Missouri State University

Campbell University

University of Michigan - Ann Arbor

Idaho State University

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.

01:39

Evaluate the integral.…

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01:12

this problem is from Chapter seven section to Problem number thirty nine in the book Calculus Early Transcendental Sze. In addition by James Store, here we have a indefinite integral of Kosi Can FX. So one way to proceed is to multiply this coast, seeking of eggs and divided by the same number. So we're not really changing the immigrant, So let's multiply and divide by Kosi Can IX plus Co Tension X Again? We haven't changed the inner rule because all we have done is multiplied co sickened by one. So here for the next step, you could multiply out that numerator and we get Kosi can squared of X plus Kosi can of X Times co attention of X and the denominator remains the same. Okay, for our next step, let's apply u substitution. Let's take you to be the denominator costigan of X plus contention of X so that do you becomes negative. Kosi can of X time's cozy in genetics minus Kosi can square of X, the ex equivalently negative Do you is our room writer because he can contention. So if we apply this u substitution, we have negative in Rule one over. You do you so we can use the power rule. Here we get a negative natural log of the absolute value of you, plus our constant of integration. See? And finally we convey back substitute you using our e substitution to have natural negative natural algorithm of absolute value. Of course he can of eggs, plus coach, engine of X plus he And that's our answer.

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