Question

Integrate x^n by ab-initio method​

Answer 1

The derivative of f(x)=x2 was found to be f'(x)=2x. Here, the derivatives of higher powers of x shall be investigate to demonstrate a pattern. Note in the algebra shown below, Pascal's triangle is used to expand powers of (x+h)n.

First Principles Differentiation of x3

The function f(x)=x3 is an antisymmetic function since f(x)=-f(-x), one can substitute x with some values to demonstrate this e.g. f(2)=8 and f(-2)=-8, therefore f(2)=-f(-2). A secant line passes through the points A(x,x3) and B(x+h,(x+h)3). f'(x) is found by taking the limit h → 0.