# Fun with Integration

**Calculus**Level 4

Solve your way through the following steps and submit the result of (c) as your answer.

(a) First, prove that for all values of \(n\in\mathbb{R}\)

\[\int _{ 0 }^{ 1 }{ (n^{ 2 }x^{ n }+2nx^{ n }+x^{ n }-1) } dx\equiv n.\]

The logical reasoning behind this proof will help in parts (b) and (c).

(b) Sketch, on the same axes, the curves representing \(f(x)=16x^3-1\) and \(x=1.\)

(c) Express \(f(x)\) in the form \( x^n (n+1)^2-1,\) and hence, without integrating or making use of any approximations, state the area enclosed between the two curves and the coordinate axes.

**Note:** This problem can easily be worked out with a calculator, online maths website or by integrating directly. But that will take all the fun out of it! The question is based on the typical calculus A-level syllabus. Please follow the rules and arrive to the answer by following all the below steps chronologically.