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Use the Substitution Formula in Theorem 7 to evaluate the integrals.$$\int_{\pi / 6}^{\pi / 4} \frac{\csc ^{2} x d x}{1+(\cot x)^{2}}$$

$ \frac{ \pi}{12} $

Calculus 1 / AB

Chapter 5

Integrals

Section 6

Definite Integral Substitutions and the Area Between Curves

Integration

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we want to use a substitution for me in the room seven to help us evaluate this integral here. Now, the way they have this written, it seems like they're leading us tow. Want to use co tangent of X to be you? Because the derivative of co tangent X should give us the differential negative. Cosi get squared X d x. If we just go out and multiply top to bottom by Daggett one or just multiplied by negative one twice. That gives us exactly d'you. Let's go ahead and plug that ends. We have d'you over one plus u squared and we know integrating that should just give us universe tangent. So let's figure out what our new bounds our first, though. All right, So when you is equal to or not, when you believe X is equal to pi forth, then you is going to be co tangent a pie fourth, which is just one start profound, this one. But what about our lower bound? So that's gonna be you is even too. So I have to think about this one. So it's co sign of high six over sign of pie sixth, cause that's just the definition of co Tanja and coast on the pie six is Route three over to Sign is going to be one over to sew the two scouts. Also, we're just left with fruit. Three are lower bound is route three. Let's go ahead and into right this. So we already said that should give us park tangent. So is negative. Tanja Inverse of you evaluated from 3 to 1. So we have the negative on the outside now plugging in one. So tangent of theta is equal to one when data is pie fourth Then we subtract off where we plug in room three and tangent of theta is equal to route three at pie Third So then pie fourth minus pi Third is negative 1/12 multiplied by the negative And we just left with pie 12th

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The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

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