I want to find the definite integral shown below, but I'm not quite sure where to start. The fastest solution apparently involved some sort of change of variables, but I can't quite find a substitution that seems to work. $$\int_0^{π/2} \frac {\cos^{2022}(x)}{\cos^{2022}(x)+\sin^{2022}(x)}\mathrm dx$$
Asked
Active
Viewed 66 times
0
-
There is in fact a very convenient change of variables: as a hint, try taking advantage of the symmetry of the denominator and the relationship between sine and cosine (the bounds are also important) – Stephen Donovan May 16 '22 at 22:15
-
Any more hints? It's been a long day and my brain seems pretty asleep right now. – Divik Verma May 16 '22 at 22:19
-
Try the change $y=\pi/2-x$ and see what you get. Once you do this change try to relate the original integral with the new one. – Marcos May 16 '22 at 22:20
-
From the denominator it can be very easy to get stuck in tunnel vision trying to do Pythagorean stuff, but the key identity is the other fundamental right angle trig identity – Stephen Donovan May 16 '22 at 22:20
-
That’s lovely. Thank you all for your help. – Divik Verma May 16 '22 at 22:45