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+13 votes

$\int^{\pi/4}_0 (1-\tan x)/(1+\tan x)\,dx $

  1. $0$
  2. $1$  
  3. $ln 2$
  4. $1/2 ln 2$
in Calculus by Boss (17.5k points)
edited by | 1.6k views

2 Answers

+37 votes
Best answer
Let $\displaystyle I = \int_{0}^{\frac{\pi}{4}}\frac{1-\tan x}{1+\tan x}dx =  \int_{0}^{\frac{\pi}{4}}\frac{\cos x-\sin x}{\cos x+\sin x}dx$

Now put $\cos x+\sin x=t\;,$ Then $\left(-\sin x+\cos x\right)dx = dt$ and changing limit

So we get $\displaystyle I = \int_{1}^{\sqrt{2}}\frac{1}{t}dt = \left[\ln t\right] = \ln(\sqrt{2}) = \frac{\ln 2}{2}$

Correct Answer: $D$
by Junior (521 points)
edited by
By using tan(45-x) i am getting


In my answer - sign is also there.

Please correct me..
I m also getting minus sign..
No. I am not getting any minus sign even doing it by tan(pi/4-x) .

yes , sandeep , u r correct.. I did  some mistake.. now got it..

no minus sign would not come
+2 votes

$\int_{0}^{\frac{\pi}{4}} \dfrac{1 - \tan x}{1 + \tan x}\\ \\ = \int_{0}^{\frac{\pi}{4}} \dfrac{\cos x - \sin x}{\cos x + \sin x}\\ \text{ Multiply and divide by cos(x)-sin(x)}\\ = \int_{0}^{\frac{\pi}{4}} \dfrac{1-2\cos x\sin x}{\cos 2x}\\ \int_{0}^{\frac{\pi}{4}} \dfrac{1 - \tan x}{1 + \tan x}\\ \\ = \int_{0}^{\frac{\pi}{4}} \dfrac{\cos x - \sin x}{\cos x + \sin x}$

Answer is D.

by Active (4.2k points)
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