Evaluate $\int^{\pi/2}_0 \frac 1 {1+\tan^\sqrt2x}dx$.

Question

Evaluate $\displaystyle \int^{\pi/2}_0 \frac 1 {1+\tan^\sqrt2x}dx$.

I tried many different methods but all failed, and I start suspecting that I can't do it in normal way (i.e. find the indefinite integral first, then substitute that bounds). I have no idea what I should do. Please give some idea!!

score 1 · Accepted Answer · answered Jan 16 '14 at 10:35

1

HINT:

Use $$\displaystyle\int_a^bf(x)dx=\int_a^bf(a+b-x)dx,$$

So, $\displaystyle I+I=\int_a^bf(x)dx+\int_a^bf(a+b-x)dx=\int_a^b\left[f(x)+f(a+b-x)\right]dx$

answered Jan 16 '14 at 10:35

lab bhattacharjee

274,582

Could you go further for me ? Thanks and cheers. – Claude Leibovici Jan 16 '14 at 10:37
@ClaudeLeibovici, http://math.stackexchange.com/questions/628295/how-to-evaluate-int-0-frac-pi2-fracdx1-tan2013x/628296#628296 and http://math.stackexchange.com/questions/439851/evaluate-the-integral-int-frac-pi2-0-frac-sin3x-sin3x-cos3xdx/439856#439856 – lab bhattacharjee Jan 16 '14 at 10:38
Thanks for that ! It is really beautiful. – Claude Leibovici Jan 16 '14 at 10:41
@ClaudeLeibovici, My pleasure. Sorry, I've found those after I've answered this. This Question can be marked as duplicate – lab bhattacharjee Jan 16 '14 at 10:50

Evaluate $\int^{\pi/2}_0 \frac 1 {1+\tan^\sqrt2x}dx$.

1 Answers1