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Numerade Educator

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Problem 37 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int^{\pi/4}_{0} \frac{1 + \cos^2 \theta}{\cos^2 \theta} \,d\theta $

Answer

$$1+\frac{\pi}{4}$$

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Video Transcript

So here it might be helpful to split this term into. So we can write see pie. I can't write pie 54 So you can write first term as one over costa and squared fair plus Co sounds great of theater. Overcoat sounds great of data. So this is just the one and so one of the coast and square theater to. That's right. This first that will just be secret squared theater plus one again to put these parentheses. And so if we look to the table of indefinite air girls, we can see that sequence squared. Theta is just equal to the integral of secret squared. The theater will be tangent theater. So here what's right tangent theater A derivative of one will be theta. A profound pi over four. Lower bound of zero. It's not about you. Right? Tangent of pipe before Plus pi over four- Attention of zero plus zero. So tangent of pie before This will be one plus pi over four. My extension of 00 plus zero. Also zero. So he gets plus one plus pi over four.