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 ..

We have: $$ \begin{align} 1^1&=1 &&& 0^1&=0 \\ 1^0&=1 &&& 0^0&=1 \end{align} $$ Are you surprised by $0^0=1$? Proof. To prove this, we consider $$ \begin{align} y=x^x \end{align} $$ and take the limit $x\rightarrow0$. It is not straightforward to do this directly with $x^x$ so we take the (natural or any) logarithm on both sides: $$ \begin{align} \ln y=\ln x^x=x\ln x \end{align} $$ Let $x=e^{-n}..

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..