Question

By using the properties of definite integrals, evaluate the integral: \(∫_0^{\frac \pi4} log(1+tanx)dx\)

Answer 1

Let \(I\) =\(∫_0^{\frac {\pi}{4}} log(1+tan x)dx\)        ...(1)

∴\(I\)= \(∫_0^{\frac {\pi}{4}} log[1+tan (\frac \pi4-x)]dx\)                    \([∫_0^af(x) dx = ∫_0^aƒ(a-x)dx]\)

⇒\(I\) = \(∫_0^{\frac  {\pi}{4}} log[1+\frac {tan\frac \pi4-tan \ x}{1+tan \frac \pi4.tan\ x}]dx\)

⇒\(I\) = \(∫_0^{\frac  {\pi}{4}} log[1+\frac {1-tan \ x}{1+tan\ x}]dx\)

⇒\(I\) = \(∫_0^{\frac  {\pi}{4}} log[\frac {2}{1+tan\ x}]dx\)

⇒\(I\) = \(∫_0^{\frac  {\pi}{4}} log\ 2\ dx - ∫_0^{\frac  {\pi}{4}} log\ (1+tan\ x)]dx\)

⇒\(I\) = \(∫_0^{\frac  {\pi}{4}} log\ 2\ dx-I\)                 [From(1)]

⇒\(2I\) = \([x.log\ 2]_0^{\frac {\pi}{4}}\)

⇒\(2I\) = \(\frac {\pi}{4}log\ 2\)

⇒I = \(\frac {\pi}{8}log\ 2\)