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Simplify i⁴?

We know that i = -1^1/2

Therefore i²=-1

As i⁴=i²xi² 

Then i⁴=-1x-1=1

C + iS = (acos(x) + a^2cos(2x) + a^3cos(3x) + ...) + i( asin(x) + a^2sin(2x) + a^3sin(3x) + ...)

= a(cos(x) + isin(x)) + a^2(cos(2x) + isin(2x)) + a^3(cos(3x) + ...

For n = 1, the sum is given by (1/2)(1)(1+1), which gives 1, the expected result. We now assume that the statement is true for some k. If we look at k+1, the sum is given by 1 + 2 + ... + k + (k+1). Since...

xln(x) - x

Some of these examiners quite like asking students to find the derivative of an inverse trig or hyperbolic function to try and catch someone off guard. The best way to approach these is to first multiply ...