1. Write z = 1 + i in polar form. 2. Using Euler\'s formula, show that f(x) = e^

Question

1. Write z = 1 + i in polar form.
2. Using Euler's formula, show that f(x) = e^ix is periodic with period 2p.
3. Calculate the interngal from 0 to 2p e^inx dx where n is an integer.
4. Calculate the interngal from 0 to 2p cos^8 (x) dx by writing cos(x) = 1/2 (e^ix + e^-ix).
Hint: Use the previous result in Problem 3 to significantly reduce the amount of work.
5. Calculate the interngal from 0 to 2p cos^n (x) dx where n is any odd integer.
Hint: The result follows immediately if you use the ideas of the previous problem. Alternatively, try the substitution x go x - p

Explanation / Answer

3)exdxdx from 0 to 2= ex^2/2dx from 0 to 2

=e(4^2)/2dx from 0 to 2

=e(4^2)/2[2-0]

=2e(4^2)/2

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