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This integral must be done using integration by parts. Therefore, we set u=ln(x) and dv=dx, which gives du=1/x and v=x.
Then, using the integration by parts formula the integral now equals x*ln(x)-i...

Two distinct real roots means that we can use b^2-4ac>0 relationship for any ax^2+bx+c equation. Apply the above gives, 4k^2 - 42(k+2)>0 Simplifying gives, k^2 - 2k -4 >0

To find the stationary points of a function we must first differentiate the function. The derivative tells us what the gradient of the function is at a given point along ...

y = (2x -3)^3

y = (2x)^3 + 3.((2x)^2)(-3) + 3.(2x).(-3)^2 + (-3)^2 using Pascal's Triangle.

y = 8x^3 - 36x^2 + 54x - 27 

dy/dx = 24x^2 - 72x + 54

at point (1,-1); dy/dx = 24 -7...

There is an initial subtle difficulty to this question, and it highlights understanding of the relationship between natural logarithms and the exponential function. One of the ways to solve this question,...