₀^{π/4}(1+ x)dx - NCERT Class XII Mathematics Chapter -

NCERT Class XII Mathematics Chapter - - Solutions

Exercise - 7.11
By using the properties of definite integrals, evaluate the integrals in Exercises 208 to 225.

Question : 214 of 271

Marks: +1, -0

\int\limits_{0}^{\pi/4}\log(1+\tan x)dx

Solution:

Let

I=\int\limits_{0}^{\pi/4}\log(1+\tan x)dx

...(i)

Also,

\int\limits_{0}^{\pi/4}\log[1+\tan(\pi/4-x)]dx

\left[\because \int\limits_{0}^{a}f(x)dx=\int\limits_{0}^{a}f(a-x)dx\right]

I=\int\limits_{0}^{\pi/4}\log\left(1+\frac{1-\tan x}{1+\tan x}\right)dx

=\int\limits_{0}^{\pi/4}\log\left(\frac{2}{1+\tan x}\right)dx

=\int\limits_{0}^{\pi/4}\log2\,dx-\int\limits_{0}^{\pi/4}\log(1+\tan x)dx

=\log 2\int\limits_{0}^{\pi/4}1\,dx-I

2I=\left[\log x\right]_{0}^{\pi/4}=(\log 2)(\pi/4-0)

\Rightarrow I=\frac{\pi}{8}\log 2

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