Consider the function f(x)=x^ 5 1-x^ 6 , 0 x<1

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Consider the function

f(x)=x^{5}\sqrt{1-x^{6}}, \qquad 0\le x<1.

Answer the following: (a) Part 1: (i) Show that the integral

\displaystyle I=\int x^{5}\sqrt{1-x^{6}}\,dx

can be rewritten in terms of a single variable raised to a power. (b) Part 2: (ii) Hence evaluate

I

in its simplest form. (c) Part 3: (iii) Using the result of part (ii), find the exact value of

\displaystyle \int_{0}^{\,\frac{1}{\sqrt[6]{2}}} x^{5}\sqrt{1-x^{6}}\,dx