Further Pure Mathematics WFM01: INTEGRATTION Topical Past Paper Questions

$$ y=\operatorname{artanh}\left(\frac{x}{\sqrt{1+x^{2}}}\right) $$ (a) Show that $$ \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{\sqrt{1+x^{2}}} $$ (b) Hence, by integrating the result in part (a), prove that $$ \operatorname{artanh}\left(\frac{x}{\sqrt{1+x^{2}}}\right)=\operatorname…

Figure 1 Figure 1 shows a sketch of the curve with equation $$ y=\cos2x\quad0\leqslant x\leqslant\frac{\pi}{4} $$ The curve is rotated through $ 2\pi $ radians about the x-axis. (a) Show that the area of the curved surface generated is given by $$ S=2\pi\int_{0}^{\frac{\pi}{4}}\c…

$$ I_{_{n}}=\int_{0}^{k}x^{n}(k-x)^{\frac{1}{2}}\mathrm{d}x\quad n\geqslant0 $$ where k is a positive constant. (a) Show that $$ I_{n}=\frac{2kn}{3+2n}I_{n-1}\quad n\geqslant1 $$ Given that $$ \int_{0}^{k}x^{2}(k-x)^{\frac{1}{2}}\mathrm{d}x=\frac{9\sqrt{3}}{280} $$ (b) use the re…