Cambridge Maths Academy

수학 모음 (Maths collection) 전체보기 This post has been partly motivated by P2 §11.6 Integration by parts CP2 §6.5 Integrating hyperbolic functions CP2 §7.1 First-order differential equations (obtaining the particular integrals using the integrating factor) Question. Derive the following results. $$ \begin{align} \textrm{(a)} &&& I_1(a,b) = \int \textrm e^{ ax } \cos b x \, \textrm dx = \frac{ \textrm ..

수학 모음 (Maths collection) 전체보기 Question. Let $$ \begin{align} I_n = \int \sin^n(ax) \, \textrm{d}x \end{align} $$ (a) Show that $$ \begin{align} I_n = -\frac1{an} \sin^{n-1}(ax) \cos(ax) + \frac{n-1}{n}I_{n-2} \end{align} $$ Let $$ \begin{align} J_n = \int_0^{\frac{\pi}{2}} \sin^nx \, \textrm{d}x \end{align} $$ (b) Hence, or otherwise, show that $$ \begin{align} J_n = \frac{n-1}{n}J_{n-2} \end{al..

수학 모음 (Maths collection) 전체보기 Question. Evaluate the following integrals: $$ \begin{align} {\rm (a)}&& &\int\frac{1}{\sqrt{1-3x^2}}\,\textrm{d}x \\ {\rm (b)}&& &\int\frac{x}{4x^2+8x+13}\,\textrm{d}x \\ {\rm (c)}&& &\int_0^1\arcsin x\,\textrm{d}x \end{align} $$ Solution. (a) By substitution, $$ \begin{align} x&=\frac1{\sqrt{3}}\sin u \\ \Rightarrow\quad \textrm{d}x&=\frac1{\sqrt{3}}\cos u\,\textr..