integral from 0 to pi/2 of sin^1000 xdx/sin^1000 x+cos^1000 x is equal to:
pi/4; pi/2; 1; 1000.
(pls. tell wch formula to be applied..)

Question

integral from 0 to pi/2 of sin^1000 xdx/sin^1000 x+cos^1000 x is
equal to:
pi/4; pi/2; 1; 1000
(pls tell wch formula to be applied )

Varun.Rawat · Accepted Answer

Let I = ∫0π/2sin1000xsin1000x + cos1000x dx    
1⇒I = ∫0π/2sin1000π2 - xsin1000π2 - x + cos1000π2 - x dx  ⇒I = ∫0π/2cos1000xcos1000x +sin1000x dx    
2Adding 1 and 2, we get     2I = ∫0π/2sin1000xsin1000x + cos1000x dx  + ∫0π/2cos1000xcos1000x +sin1000x dx    ⇒2I = ∫0π/2sin1000x + cos1000xsin1000x + cos1000x dx ⇒2I = ∫0π/2dx⇒2I = x0π/2⇒2I = π2 - 0⇒I = π4

integral from 0 to pi/2 of sin^1000 xdx/sin^1000 x+cos^1000 x is equal to: pi/4; pi/2; 1; 1000. (pls. tell wch formula to be applied..)

integral from 0 to pi/2 of sin^1000 xdx/sin^1000 x+cos^1000 x is equal to:
pi/4; pi/2; 1; 1000.
(pls. tell wch formula to be applied..)