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Evaluate the integral.

$ \displaystyle \int x \tan^2 x dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

Missouri State University

Campbell University

Baylor University

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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Evaluate the integral.…

here we have a mineral of X times tangent squared of X. Let's go ahead and use integration. My parts Let's take unit B X so that do you is the ex. Then we're left over with d V equals Tan Squared X, which we can also write using our pathology and identities seek and squared minus one, and doing writing it this way makes it easier to find the So it's a fine B which is integrate so the integral C can't square is tangent and the integral of one is X. So here, using parts U V minus integral medio we have ext time San Genetics minus X squared. So this is you need minus inaugural video. So there's our you ve and over here underlined is in a girl video. All right, for the next step, let's just rewrite this u b we have x ten x minus, X squared minus and our world So it's quite and bright. This is two separate minerals and then here we have a double negative. So have a plus into a Rolex I couldn't separate. Some are scratch work. So now we know how to evaluate these inner girls. Xed annex minus X square so anti derivative of tangent is natural law of seeking absolute value of seeking. You could either attain this result from your table or here you can go ahead and use it. Use up. So first you can right tangent equals sign of our coastline. And then you could do it, you substitution. That's another way. Toe. Evaluate this anti derivative and dance I derivative of X. We're exclude over, too, and let's add our constancy. So the last thing we can do here is combined like terms. We have a negative X square plus X squared over two sorts commandos. So doing so we have negative X squared over two plus X tan X minus Ellen. Absolute value seeking a Vicks plus he and there's our answer.

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