Let $f’(x) = \frac{{3{x^2}}}{{{{({x^3} + 1)}^5}}}$. Given that $f(0) = 1$, find $f(x)$.

valid approach     (M1)

eg$\,\,\,\,\,$$\int {f'{\text{d}}x,{\text{ }}\int {\frac{{3{x^2}}}{{{{({x^3} + 1)}^5}}}{\text{d}}x} }$

correct integration by substitution/inspection     A2

eg$\,\,\,\,\,$$f(x) =  - \frac{1}{4}{({x^3} + 1)^{ - 4}} + c,{\text{ }}\frac{{ - 1}}{{4{{({x^3} + 1)}^4}}}$

correct substitution into their integrated function (must include $c$)     M1

eg$\,\,\,\,\,$$1 = \frac{{ - 1}}{{4{{({0^3} + 1)}^4}}} + c,{\text{ }} - \frac{1}{4} + c = 1$

Note:     Award M0 if candidates substitute into $f’$ or $f’’$.

$c = \frac{5}{4}$     (A1)

$f(x) =  - \frac{1}{4}{({x^3} + 1)^{ - 4}} + \frac{5}{4}{\text{ }}\left( { = \frac{{ - 1}}{{4{{({x^3} + 1)}^4}}} + \frac{5}{4},{\text{ }}\frac{{5{{({x^3} + 1)}^4} - 1}}{{4{{({x^3} + 1)}^4}}}} \right)$     A1     N4

[6 marks]