# math

Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit.

Consider the definite integral from pi/6 to pi/2 cos(z)/sin5(z)dz

Then the most appropriate substitution to simplify this integral is
u =

Then dz=f(z)du where
f(z) =

After making the substitution and simplifying we obtain the integral abg(u)du where
g(u) =
a =
b =

This definite integral has value =

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1. I assume you mean

∫cos(z)/sin^5(z) dz
If u = sin(z), then du = cos(z) dz and we have
∫u^-5 du
= -1/4 u^-4

Now the limits of integration become [1/2,1] because u = sin(z)

Now it's easy to evaluate:

(-1/4 * 1^-4)-(-1/4 * 2^4) = 15/4

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