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JH
Numerade Educator

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Problem 82 Hard Difficulty

Evaluate the integral.

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

Answer

$\frac{1}{2} \tan ^{-1}\left(2 \sin ^{2} x-1\right)+C$

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Video Transcript

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.