Question

Evaluate : \[\int\limits_0^\frac{\pi}{4} \tan x dx\] .

Answer 1

I = \[\int\limits_0^\frac{\pi}{4} tan \ x \ dx\]

= \[\left[ \log\left| secx \right| \right]_0^\frac{\pi}{4}\]

= \[\log\left| sec\frac{\pi}{4} \right| - \log\left| sec0 \right|\]

= \[\log\left| \sqrt{2} \right| - \log1\]

= \[\frac{1}{2}\log2\]

∴ \[\int\limits_0^\frac{\pi}{4} tan\ x \ d x\] = \[\frac{1}{2}\log2\]