# Hello world!

Welcome to Mathstoon!

Problem 1: Evaluate $\int e^{2x} \,dx$

Let $2x=t$.

$\int \frac{\tan x}{\cot x} dx$

$=\int \tan^2 x \ dx$

$= \int (\sec^2 x-1)\ dx$

Problem 2: Show that $\log_3 \log_2 8=1$

Solution:

At first, we calculate $\log_2 8$.

Now, $\log_2 8=\log_2 2^3$

$=3 \log_2 2 \quad$ $[\because \log_a b^k=k \log_a b]$

$=3 \cdot 1 \quad$ $[\because \log_a a=1]$

$=3$

$\therefore$ LHS $= \log_3 \log_2 8$

$=\log_3 3=1$ (proved)

 ${\dfrac{d}{dx}(x^n)=\dfrac{d}{dx}(f(x))} {=\lim\limits_{h \to 0}\dfrac{f(x+h)-f(x)}{h}} {=\lim\limits_{h \to 0}\dfrac{(x+h)^n-x^n}{h}}$ [Let $z=x+h.$ Then $z \to x$ as $h \to 0$] $=\lim\limits_{z \to x} \dfrac{z^n-x^n}{z-x}=nx^{n-1}$

### 1 thought on “Hello world!”

1. Hi, this is a comment.
To get started with moderating, editing, and deleting comments, please visit the Comments screen in the dashboard.
Commenter avatars come from Gravatar.