∫(0 to pi/2) min(sinx,cosx)dx,equals to:

1)2-sqrt2
2)2+sqrt2
3)2sqrt2
4)sqrt2

Split this integral at x = π/4 (which is where sin x = cos x).

∫(x = 0 to π/2) min(sin x, cos x) dx
= ∫(x = 0 to π/4) min(sin x, cos x) dx + ∫(x = π/4 to π/2) min(sin x, cos x) dx
= ∫(x = 0 to π/4) sin x dx + ∫(x = π/4 to π/2) cos x dx
= (-cos x) {for x = 0 to π/4} + (sin x) {for x = π/4 to π/2}
= (-√2/2 - (-1)) + (1 - √2/2)
= 2 - √2.

I hope this helps!