Standard +0.3 This is a straightforward substitution question with clear guidance (substitution given explicitly). Students must find du/dx = 4x³, rearrange to get x⁴ in terms of u, change limits, and integrate. The algebra is slightly more involved than basic substitution due to the x⁷ term requiring factoring as x⁴·x³, but this is a standard C3 technique. The final answer form requiring logarithms suggests partial fractions or a log result from integration, making it slightly above average difficulty but well within expected C3 scope.
6 Use the substitution \(u = x ^ { 4 } + 2\) to find the value of \(\int _ { 0 } ^ { 1 } \frac { x ^ { 7 } } { \left( x ^ { 4 } + 2 \right) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(p \ln q + r\), where \(p , q\) and \(r\) are rational numbers.
6 Use the substitution $u = x ^ { 4 } + 2$ to find the value of $\int _ { 0 } ^ { 1 } \frac { x ^ { 7 } } { \left( x ^ { 4 } + 2 \right) ^ { 2 } } \mathrm {~d} x$, giving your answer in the form $p \ln q + r$, where $p , q$ and $r$ are rational numbers.
\hfill \mbox{\textit{AQA C3 2012 Q6 [6]}}