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Evaluate the integral.

$ \displaystyle \int \frac{\sin x \cos x}{\sin^4 x + \cos^4 x}\ dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

Integration Techniques

Oregon State University

Harvey Mudd College

University of Nottingham

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.

02:20

Evaluate the integral.…

12:49

01:25

00:37

Evaluate the integrals.

01:42

Let's start here by dividing top and bottom. Bye. Let's do co sign to the fourth power. So just rewriting this divide this by co signed to the fourth and to make sure we don't change anything. We also do this on the bottom. Co signs in the forest and then coast since the fourth years. Well, and then let's go ahead and rewrite this. So we have sign eggs and then times one over cosign cute and then down here, that's a one. And then this is just tangent to the fourth. So let's re write this numerator tonight. A signed over co sign and then times one over coastlines where don't give us attention, c can swear. And then let's do it. Ten square x Squyres saw stance of the four plus one. And then finally, will you sup here? So we should go ahead and take that to be our substitution. So it's going on the next page to write the rest out, So we'LL have So let's let's rewrite what we had First ten sequence where Tan Square squared plus one and we chose unit B tanz Clear t That should be our let's actually choose this to be Are you not the one we wrote on the previous page? The reason for doing this is that we'LL get one extra tangent here Before we get this, he can't square. I should be doing exes here. Sorry. So let me actually scratch all that, Do you Over too will give us the noon. Write her back so we can write. This is one half and then do you and then use where? Plus one. Now, if you don't remember this one two tricks up new equals tan data and you will arrive at the following That's it are ten of you plus e And then we use our formula up here to re write this Stan in verse, hands squared of eggs and then at our constancy And that's the final answer.

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