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Let I = integral[(lnx)/x^2 dx] for simplicity.Firstly, we realise we must use integration by parts. This is:Integral [u(x)v'(x) dx] = u(x)v(x) - Integral[u'(x)v(x) dx]So we can see that, by letting u(x)=l...

Differentiate y with respect to x using quotient rule:y'=[(1/x)(5x^2)-(10x)(lnx)]/(lnx)^2 =[5x-10xlnx]/(lnx)^2Stationary points occur when y'=0, so when y'=0 we have:5x-10xlnx = 0x(5-10lnx)=0So x=0 or 5...

The turning point of a curve is the point at which the gradient is 0 as from there it stops rising and starts falling or vice versa. To find this we differentiate y with respect to x (dy/dx) to find the g...

We argue using the discriminant of a quadratic polynomial. For a quadratic ax^2 + bx + c=0, the discriminant D is D = b^2 - 4ac. When this quadratic has two distinct real roots, we have that D > 0. So ...

I don't understand why the function "f(x)=x^2 for all real values of x" has no inverse. Isn't sqrt(x) the inverse?
I like to think of a function a bit like a machine. It takes in a number, ...