2022 H2 Mathematics Paper 2 Question 1

Integration Techniques

Answers

23(x+2)3−4x+2+c\displaystyle \frac{2}{3} \sqrt{ (x+2)^3 } - 4\sqrt{x+2} + c

Full solutions

dudx=12x+2=12u\begin{align*} \frac{\mathrm{d}u}{\mathrm{d}x} &= \frac{1}{2\sqrt{x+2}} \\ &= \frac{1}{2u} \end{align*}
x=u2−2x = u^2 - 2
∫xx+2  dx=∫u2−2u  (2u)  du=2u2−4  du=23u3−4u+c=23(x+2)3−4x+2+c  ■\begin{align*} & \int \frac{x}{\sqrt{x+2}} \; \mathrm{d}x \\ &= \int \frac{u^2 - 2}{u} \; (2u) \; \mathrm{d}u \\ &= 2u^2 - 4 \; \mathrm{d}u \\ &= \frac{2}{3} u^3 - 4u + c \\ &= \frac{2}{3} \sqrt{ (x+2)^3 } - 4\sqrt{x+2} + c \; \blacksquare \end{align*}

Question Commentary

A relative direct integration by substitution question. Make sure to differentiate the given substitution first, and replace dx{\mathrm{d}x} with du{\mathrm{d}u} via the chain rule. Also don't forget to change the final answer back to the original variable x.{x.}