Question 3 - A-Level Maths

Exam Board	OCR MEI
Module	C3 (Core Mathematics 3)
Topic	Integration by Substitution

Use the substitution $u = 1 + x$ to show that $$\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { 1 + x } \mathrm {~d} x = \int _ { a } ^ { b } \left( u ^ { 2 } - 3 u + 3 - \frac { 1 } { u } \right) \mathrm { d } u$$ where $a$ and $b$ are to be found.
Hence evaluate $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { 1 + x } \mathrm {~d} x$, giving your answer in exact form. Fig. 8 shows the curve $y = x ^ { 2 } \ln ( 1 + x )$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d1206ce8-7716-4205-b98e-664e7ead8a25-3_830_806_907_706} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$. Verify that the origin is a stationary point of the curve.
Using integration by parts, and the result of part (i), find the exact area enclosed by the curve $y = x ^ { 2 } \ln ( 1 + x )$, the $x$-axis and the line $x = 1$.