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let f(x) = ax^3 + ax^2 + ax - 42if x-2 is a factor of f(x), then by remainder theorem, f(2) = 0and:a(2)^3 + a(2)^2 + a(2) -42 = 08a + 4a + 2a - 42 = 014a = 42a = 3

u=lnx du/dx = 1/x dv/dx=x^-2 v= -1/x =uv - (integral of)vdu/dx (-lnx)/x + integral of x^-2 =(-lnx)/x - 1/x +c

First we need to factorise the denominator into (x-1)(x+1). Now we identify that 1/(x-1)(x+1) is equivalent to A/(x-1) + B/(x+1) and now we solve to find the values of A and B. By simplifying the equivalent ...

logx(25) = log5(x)2logx(5) = log5(x)2/log5(x) = log5(x)2 = (log5(x))^2sqrt(2) = log5(x)x = 5^sqrt(2)

Firstly, look at the number in front of the letter 'p'. This is 5p and -3p. By ignoring the 'p', as they both have it, do 5-3 which is 2 and so that part of the answer will be 2p. Then look at the letter 'q'...