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we must integrate (4cos⁴ x-4cos² x+1)^1/2 factorise first ((2cos²x-1)²)^1/2this becomes (2cos²x-1)which equals cos 2xthe integral...

First we start off by seeing that we are multiplying together two functions both containing x, so we want to apply the product rule. As we know the product rule is (f(x)g(x))'=f(x)g'(x)+f'(x)g(x) so we ca...

First multiply top and bottom by conjugate of denominator, (7-3√5), and expand(3 + 2√5)(7 - 3√5)/(7 + 3√5)(7 - 3√5)(21 + 14√5 - 9√5 - 30)/(49 + 3√5 - 3√5 - 45)Simplify top and botton(-9 + 5√5)/4Write in r...

(1+6x)^3 = 1+3(6x) +(3)(2)(36x^2)/2 + (3)(2)(1)(216x^3)/6
= 1+18x+108x^2 + 216x^3

Question is asking for gradient at x = 0, y = ln2. e^2y = 5 - e^-x. Differentiation with respect to x: 2e^2y * dy/dx = e^-x . dy/dx = e^-x / 2e^2y. At x = 0, y = ln2 ~ dy/dx = e^0 / 2e^2ln2 = 1 / 2e^ln4 =...