klimenkov one year ago $\int_{-\frac{\pi}2}^{\frac{\pi}2} \frac{\sin^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx$

1. ganeshie8

even/odd stuff ha ?

2. BSwan

2014 is order of derivative ?

3. hartnn

$$\Large \int_a^b f(x) dx= \int_a^bf(a+b-x)dx$$ use this!

4. klimenkov

@BSwan 2014 is the power.

5. hartnn

after you have proved that your function is even

6. BSwan

aha cool :)

7. hartnn

$$\Large \int_{-a}^a f(x) dx = 2\int_0^a f(x)dx$$ if f(x) is even function

8. hartnn

did u get what to do ?

9. klimenkov

$2\int\limits_0^{\frac{\pi}2} \frac{\sin^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx$What's next?

10. hartnn

$$\Large \int_a^b f(x) dx= \int_a^bf(a+b-x)dx$$ replace x by pi/2 - x

11. hartnn

$$I = 2\int\limits_0^{\frac{\pi}2} \frac{\sin^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx ... ... (A)$$ just giving a label, to be used later

12. klimenkov

$2 \int\limits_0^{\frac{\pi}2} \frac{\cos^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx$

13. klimenkov

Oops...

14. hartnn

$$I = 2 \int\limits_0^{\frac{\pi}2} \frac{\cos^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx ... ... (B)$$ Add (A) and (B)

15. klimenkov

$I = -2 \int\limits_0^{\frac{\pi}2} \frac{\cos^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx$

16. hartnn

what ? why negative ?

17. hartnn

dx remains as dx

18. hartnn

i am NOT doing any substitution

19. klimenkov

Yeah, everything is okay. My fault. $I + I = 2\int_0^{\frac\pi2}dx,$$I=\frac\pi2.$Very nice, thank you.

20. hartnn

welcome ^_^