2008 H2 Mathematics Paper 1 Question 5

Integration Techniques

Answers

(i)

π9{\frac{\pi}{9}}

(ii)

nen+1+1(n+1)2{\frac{n\mathrm{e}^{n+1}+1}{(n+1)^2}}

Full solutions

(i)

∫01311+9x2  dx=19∫013119+x2  dx=19[113tan⁡−1(x13)]013=[13tan⁡−13x]013=π9  ■\begin{align*} & \int_0^{\frac{1}{\sqrt{3}}} \frac{1}{1 + 9 x^2} \; \mathrm{d}x \\ & = \frac{1}{9} \int_0^{\frac{1}{\sqrt{3}}} \frac{1}{\frac{1}{9} + x^2} \; \mathrm{d}x \\ & = \frac{1}{9} \left[ \frac{1}{\frac{1}{3}} \tan^{-1} \left(\frac{x}{\frac{1}{3}}\right) \right]_0^{\frac{1}{\sqrt{3}}} \\ & = \left[ \frac{1}{3}\tan^{-1} 3 x \right]_0^{\frac{1}{\sqrt{3}}} \\ & = \frac{\pi}{9} \; \blacksquare \\ \end{align*}

(ii)

∫1exnln⁡x  dx=[xn+1n+1ln⁡x]1e−∫1exn+1n+11x  dx=en+1n+1−1n+1∫1exn  dx=en+1n+1−1n+1[xn+1n+1]1e=en+1n+1−en+1(n+1)2+1(n+1)2=(n+1)en+1−en+1+1(n+1)2=nen+1+1(n+1)2  ■\begin{align*} & \int_1^\mathrm{e} x^n \ln x \; \mathrm{d}x \\ & = \left[ \frac{x^{n+1}}{n+1} \ln x \right]_1^\mathrm{e} - \int_1^\mathrm{e} \frac{x^{n+1}}{n+1} \frac{1}{x} \; \mathrm{d}x \\ & = \frac{\mathrm{e}^{n+1}}{n+1} - \frac{1}{n+1} \int_1^\mathrm{e} x^n \; \mathrm{d}x \\ & = \frac{\mathrm{e}^{n+1}}{n+1} - \frac{1}{n+1} \left[ \frac{x^{n+1}}{n+1} \right]_1^\mathrm{e} \\ & = \frac{\mathrm{e}^{n+1}}{n+1} - \frac{\mathrm{e}^{n+1}}{(n+1)^2} + \frac{1}{(n+1)^2} \\ & = \frac{(n+1)\mathrm{e}^{n+1}-\mathrm{e}^{n+1}+1}{(n+1)^2} \\ & = \frac{n\mathrm{e}^{n+1}+1}{(n+1)^2} \; \blacksquare \end{align*}