Question

$\large{\sf \red{Hello\ Brainlians\ !}}$
$\bf{Evaluate\ the\ integral\ ,}$
$\displaystyle \rm \int \dfrac{1+e^x}{1-e^x}\ dx$

Answer 1

Problem :

$\bullet\ \; \; \displaystyle \red{ \sf \int \dfrac{1+e^x}{1-e^x}\ dx }$

Solution :

$\displaystyle \sf \int \dfrac{1+e^x}{1-e^x}\ dx$

$⇒ \sf \displaystyle \sf \int \dfrac{1+e^x+e^x-e^x}{1-e^x}\ dx$

$⇒ \sf \displaystyle \sf \int \dfrac{1-e^x+2e^x}{1-e^x}\ dx$

$⇒ \sf \displaystyle \sf \int \left( \dfrac{1-e^x}{1-e^x} + \dfrac{2e^x}{1-e^x} \right) dx$

$⇒ \sf \displaystyle \sf \int \dfrac{1-e^x}{1-e^x}\ dx -2 \int \dfrac{(-e^x)}{1-e^x}\ dx$

$⇒ \sf \displaystyle \sf \int dx -2 \int \dfrac{(-e^x)}{1-e^x}\ dx$

2nd integral resembles ∫ f'(x)/f(x) dx = ㏑ f(x)

$\longrightarrow \bf x-2 \ln (1-e^x)+c$

Answer 2

Correct option is

C

Statement -I is False; Statement -II is True

I=∫π/6π/31+tanxdx

I=∫π/6π/31+tanxtanxdx

2I=6π

I=12π

Therefore, I is false and II is true.

Hence, option 'C' is correct.