Challenging +1.8 This is a Further Maths integration by parts question requiring knowledge of the arcsinh function and its derivative. While the technique is standard (integration by parts with u = arcsinh 2x), students must correctly handle the derivative of arcsinh, perform algebraic manipulation with the resulting integral, and evaluate definite integral limits involving arcsinh(2π/3). The 8-mark allocation and 'show detailed reasoning' instruction indicate substantial working, but the path is relatively clear for Further Maths students who know the standard approach to integrating inverse hyperbolic functions.
In this question you must show detailed reasoning.
Show that
$$\int_0^{\frac{\pi}{3}} \operatorname{arcsinh} 2x \, dx = \frac{2}{3} \ln 3 - \frac{1}{3}.$$ [8]
In this question you must show detailed reasoning.
Show that
$$\int_0^{\frac{\pi}{3}} \operatorname{arcsinh} 2x \, dx = \frac{2}{3} \ln 3 - \frac{1}{3}.$$ [8]
\hfill \mbox{\textit{OCR MEI Further Pure Core Q15 [8]}}