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Numerade Educator

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Problem 38 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int^{\pi/3}_{0} \frac{\sin \theta + \sin \theta \tan^2 \theta}{\sec^2 \theta} \,d\theta $

Answer

$\int_{0}^{\frac{\pi}{3}} \frac{\sin \theta+\sin \theta \tan ^{2} \theta}{\sec ^{2} \theta} d \theta=\frac{1}{2}$

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Video Transcript

All right. This is uh kind of a goofy problem. That's very doable. We're looking at 02 power three and it looks super complicated. But just stick with me like that. Sine of theta plus sine theta tangent squared of theta. All over. 2nd squared of freedom. The theater. Now, if you know your I guess we need to factor first. I just rewrite that this is equal to factoring out a sign of data in the numerator. Let's do that trick first. That's just factoring skills. You should know that from Algebra one. It's the greatest common factor. If you don't believe me distributed and you'll see it's the same thing. But in pre cal core even algebra two, we learned that one plus tangent squared is always equal to seek and squared. Um So that's a a trig identity. If you're confused, I would tell you to google it or go back to pre coke that one plus tangent squared is equal to second squared. So we can cancel those two things out. So now we're ready to just look at this and say, ok well do we know the anti derivative of sine? And we do. It's negative co sign. If you don't believe me, just you know what's the derivative of cosine? It's negative sign. So if it's negative coastline would be positive sign And that goes from 0 to Pi over three. Well now we can start plugging in your upper and lower bounds. Male switch colors again Because Pi over three is up here at this ordered pair And then zero. Is this sordid pair? 10. So as we plug in pirate three, we're looking at that value. You're looking at negative 1/2 minus. We're gonna plug in zero and for the and co signs the X coordinate. So it becomes minus negative one because the negative is still there. So really what we're looking at is negative on half plus one, Which gives me positive 1/2 is your correct answer. So hopefully you were with me on every step of the way, but this is the correct answer.