Browse Questions

# Evaluate the definite integral$\int\limits_0^\frac{\Large \pi}{\Large 4}\tan x\;dx$

Toolbox:
• (i)$\int \limits_a^b f(x)dx=F(b)- F(a)$
• (ii)$\int \tan x dx= -log |\cos x| \;or\; log|\sec x|$
Given $I=\int \limits_0^{\pi/4} \tan x dx$

On integrating we get

$\bigg[log |sec x|\bigg]^{\pi/4}_0$

On applying limits we get

$log(\sec \frac{\pi}{4}-\sec 0)$

But $\sec \frac{\pi}{4}=\sqrt 2\;and\;\sec 0=1$

Therefore $I=log|\sqrt 2-1|$

But $log a-log b=log(a/b),$ similarly

$log \frac{\sqrt 2}{1}=log \sqrt 2$

$=log 2^{1/2}$

But $log x^a=a log x,$ similarly,

We can write $\log 2^{1/2}=\frac{1}{2} log 2$

Hence $\int \limits_0^{\pi/4} \tan x dx=\frac{1}{2} log 2$