Integration Techniques: IBP: \(\cos(3\ln x)\); substitution — NJC 2025 H2 Math Prelim Paper 1

(a)

By parts with \(u=\cos(3\ln x)\), \(\mathrm{d}v=\mathrm{d}x\): \[\begin{aligned} \int\cos(3\ln x)\,\mathrm{d}x &= x\cos(3\ln x) - \int x\!\left(-\sin(3\ln x)\!\left(\frac{3}{x}\right)\right)\mathrm{d}x \\ &= x\cos(3\ln x) + 3\int\sin(3\ln x)\,\mathrm{d}x \end{aligned}\]

By parts again on \(\displaystyle\int\sin(3\ln x)\,\mathrm{d}x\): \[\begin{aligned} \int\sin(3\ln x)\,\mathrm{d}x &= x\sin(3\ln x) - \int x\cos(3\ln x)\!\left(\frac{3}{x}\right)\,\mathrm{d}x \\ &= x\sin(3\ln x) - 3\int\cos(3\ln x)\,\mathrm{d}x \end{aligned}\]

Substituting back: \[ \int\cos(3\ln x)\,\mathrm{d}x = x\cos(3\ln x)+3x\sin(3\ln x)-9\int\cos(3\ln x)\,\mathrm{d}x \]

Hence \(10\displaystyle\int\cos(3\ln x)\,\mathrm{d}x = x\cos(3\ln x)+3x\sin(3\ln x)+c\).

\(\displaystyle\int\cos(3\ln x)\,\mathrm{d}x = \dfrac{x}{10}\cos(3\ln x)+\dfrac{3x}{10}\sin(3\ln x)+C\)

(b)(i) \(\mathrm{P}(x)=1-x\): \[ \int\frac{1-x}{1-\sqrt{x}}\,\mathrm{d}x = \int\frac{(1-\sqrt{x})(1+\sqrt{x})}{1-\sqrt{x}}\,\mathrm{d}x = \int(1+\sqrt{x})\,\mathrm{d}x \]

\(I = x + \dfrac{2}{3}x^{3/2} + c\)

(b)(ii) \(\mathrm{P}(x)=1\), substitution \(u=1-\sqrt{x}\):

\(\dfrac{\mathrm{d}u}{\mathrm{d}x} = -\dfrac{1}{2\sqrt{x}} = -\dfrac{1}{2(1-u)}\), so \(\mathrm{d}x = -2(1-u)\,\mathrm{d}u\). \[\begin{aligned} \int\frac{1}{1-\sqrt{x}}\,\mathrm{d}x &= \int\frac{1}{u}(-2)(1-u)\,\mathrm{d}u = 2\int\!\left(-1+\frac{1}{u}\right)\mathrm{d}u \\ &= 2(-u+\ln|u|)+c \\ &= 2(-(1-\sqrt{x})+\ln(1-\sqrt{x}))+c \\ &= -2\sqrt{x}-2\ln(1-\sqrt{x})+C \qquad (0<x<1,\; u>0) \end{aligned}\]

\(\displaystyle\int\frac{1}{1-\sqrt{x}}\,\mathrm{d}x = -2\sqrt{x}-2\ln(1-\sqrt{x})+C\)

For \(\mathrm{P}(x)=x\): write \(x = -(1-x)+1\): \[\begin{aligned} \int\frac{x}{1-\sqrt{x}}\,\mathrm{d}x &= -\int\frac{1-x}{1-\sqrt{x}}\,\mathrm{d}x + \int\frac{1}{1-\sqrt{x}}\,\mathrm{d}x \\ &= -\!\left(x+\frac{2}{3}x^{3/2}\right) + (-2\sqrt{x}-2\ln(1-\sqrt{x}))+c \\ &= -x-\frac{2}{3}x^{3/2}-2\sqrt{x}-2\ln(1-\sqrt{x})+C \end{aligned}\]