2.19: Quotient rule, integration and inverse function

0606/21/O/N/16 — Question 8 · 11 marks

The function

f(x)

is given by

f(x) = \dfrac{3x^{3} - 1}{x^{3} + 1}

for

0 \le x \le 3

.

(i)

Show that

f'(x) = \dfrac{kx^{2}}{(x^{3} + 1)^{2}}

, where

k

is a constant to be determined.

[3]

(ii)

Find

\displaystyle\int \frac{x^{2}}{(x^{3} + 1)^{2}}\,\mathrm{d}x

and hence evaluate

\displaystyle\int_{1}^{2} \frac{x^{2}}{(x^{3} + 1)^{2}}\,\mathrm{d}x

.

[4]

(iii)

Find

f^{-1}(x)

, stating its domain.

[4]