Mathematics 7357: Integration Topical Past Paper Questions

The curve C with equation $$ y=(x^{2}-8x)\ln x $$ is defined for $x>0$ and is shown in the diagram below. The shaded region, $R$, lies below the $x$-axis and is bounded by $C$ and the $x$-axis. Show that the area of R can be written as $$ p+q\ln2 $$ where p and q are rational num…

Use the substitution $ u = x^{5} + 2 $ to show that $$ \int_{0}^{1}\frac{x^{9}}{(x^{5}+2)^{3}}\mathrm{d}x=\frac{1}{180} $$ [7 marks] ___ ___ ___ ___ ___ ___ ___ ___ ___

Find $$ \int\left(x^{2}+x^{\frac{1}{2}}\right)\mathrm{d}x $$ [2 marks] [∠ marks]