Flag Integral Calculus> Please calculate the integral limit is (0...
question mark

Please calculate the integrallimit is (0 to pi/4)∫ln(1+tanx) dx

Kawal , 10 Years ago
Grade 11
anser 1 Answers
Sumit Majumdar

Last Activity: 10 Years ago

Dear student,
We have:
J = ∫ ln { 1 + tan [ (π/4) - x ] } dx ... on [0,π/4]

...= ∫ ln { 1 + [ ( 1 - tan x ) / ( 1 + tan x ) ] } dx

...= ∫ ln { 2 / ( 1 + tan x ) } dx

...= ∫ { ( ln 2 ) - ln ( 1 + tan x ) } dx ... on [0,π/4]

...= [ ( ln 2 ) ∫ dx ] - J .... from (2)
_____________________________________

∴ J = ( ln 2 ) { x on [0,π/4] } - J

∴ 2J = ( ln 2 ) [ (π/4) - 0 ]

∴ 2J = (π/4) ( ln 2 )

∴ J = ( π/8 )· ln 2 = (0.25)π· ln 2I=\int_{0}^{\frac{\pi }{4}} ln\left ( 1+tanx \right )dx=\int_{0}^{\frac{\pi }{4}} ln\left ( 1+tan\left (\frac{\pi }{4}-x \right ) \right )dx=\int_{0}^{\frac{\pi }{4}} ln\left ( 1+ \frac{1-tan\left ( x \right )}{1+tan\left ( x \right )}\right )dx=\int_{0}^{\frac{\pi }{4}} ln\left ( \frac{2}{1+tan\left ( x \right )}\right )dx=\int_{0}^{\frac{\pi }{4}} \left (ln\left ( 2 \right ) \right -ln\left ( 1+tan\left ( x \right ) \right ))dx=\int_{0}^{\frac{\pi }{4}} \left ( ln2 \right )dx-I\Rightarrow I=\frac{1}{2}\int_{0}^{\frac{\pi }{4}} \left ( ln2 \right )dx=\frac{\pi }{8}ln2Regards
Sumit

Provide a better Answer & Earn Cool Goodies

Enter text here...
star
LIVE ONLINE CLASSES

Prepraring for the competition made easy just by live online class.

tv

Full Live Access

material

Study Material

removal

Live Doubts Solving

assignment

Daily Class Assignments


Ask a Doubt

Get your questions answered by the expert for free

Enter text here...