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Evaluate the integral.

$ \displaystyle \int \frac{1 - \tan^2 x}{\sec^2 x} dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

Oregon State University

Baylor University

Boston College

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.

03:40

02:19

Evaluate the integral.…

09:32

05:58

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01:03

00:32

Evaluate the indefinite in…

00:45

04:59

this problem is from Chapter seven, Section two. Problem forty seven In the book Calculus Early Transcendental lt's a Condition by James Door. We have an indefinite integral of one minus tan squared on the numerator divided by C can swear. So let's go to the right and make some observations. One of our protagonist identities tells us that one plus tan square to see can't square. So this means that we have tan squared, seeking squared minus one, and this means one minus hand square is one minus seek inns where minus one, which is to minus seek and square. What so we can replace under Renner with two minus, he can't square. And the interval we have seeking square still in the denominator. Let God, it's split us up into two fractions. We have to oversee cans where should be into up there and then we have seeking squared over seek and squared, which is just one. Then we could simplify this even further. This is two coasts and squared eggs minus one D X, and this can be integrated using a formula for co sign squared. But actually, in this case, you can go ahead and rewrite then a grand using the double angle formula. This is simply co sign of two works. If you didn't realize this fact than another way to proceed would have been to take this coastline square and right It is one plus coastline of two X over two. Even after doing this, you will still ended up at the same equation that we're right now. Then we know the integral this and my help here excellent might help you to do a U substitution. After doing so, we should get sign of two X over to plus he and that's our answer.

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