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Start with finding limits by setting 3x - x^2 = 0, then factorise x(3 - x) = 0. Therefore x = 0 or 3. The area is the integral of 3x - x^2 between x = 0 and 3, sub in 3 and 0 into 3(x^2)/2 - (x^3)/3, whic...

2x/[(x+1)(2x-4)] = [A/(x+1) + B/(2x-4)]x = A(x-2) + B(x+1)x = -1-1 = A(-3)A = 1/3x = 2 2 = B(3)B = 2/3 therefore.... 2x/[(x+1)(2x-4)] ...

Let us procede by induction:

First case: n=1. Then LHS (left hand side) = 1 and RHS (right hand side) = 1(1+1)/2 = 1. Therefore, we see that the statement is true for n=1.

Now, we carry out ...

x = ln (sec (2y))

The chain rule states that d/dy f (g (y)) = f'(g(y)). g'(y)

Here g(y) = sec(2y) so g'(y) = 2.sec(2y).tan(2y)

And f(y) = ln (y) so f'(y) = 1 / y

Thus dx/dy = (...

f''(x) means that we need to differentiate the function f(x) twice (f'(x) would mean we need to do it once). Differentiation means we multiply the coefficient by the power, and subtract one from the power...