Evaluate (i) `int_0^(pi//4) sin x cos x dx` (ii) `int_0^(pi//2) (1 + cos x)^(1//2) dx` (iii) `int_0^(pi//2) (1 + sin x)^(1//2) dx` (iv) `int_0^(pi//4) (1 -cos 2x)^(1//2)dx`

Evaluate (i) `int_0^(pi//4) sin x cos x dx` (ii) `int_0^(pi//2) (1 + c

1.	Evaluate (i) `int_0^(pi//4) sin x cos x dx` (ii) `int_0^(pi//2) (1 + cos x)^(1//2) dx` (iii) `int_0^(pi//2) (1 + sin x)^(1//2) dx` (iv) `int_0^(pi//4) (1 -cos 2x)^(1//2)dx`
Answer» Correct Answer - (i)`(1)/(4)` (ii) 2 (iii) 2 (iv) 0.414 (i) `int_0^(pi//4) sin x cos x dx = (1)/(2) int_0^(pi//4) sin 2x dx = (1)/(2)[(-cos 2x)/(2)]_0^(pi//4) = (1)/(4)` (ii) `overset(pi//2)underset0int(1+cosx^(1/2))dxoverset(pi//2)underset0intsqrt2cosx/2dx=sqrt2((sin(x)/(2))/((1)/(2)))=2sqrt2[sin(pi//2)/2-(sin0)/2]` `=2sqrt2[sinpi/4]=2sqrt2xx1/sqrt2=2` ` =2sqrt2[sin(pi)/(4)] = 2sqrt2xx(1)/(sqrt2) =2` (iii) `overset(pi//2)underset0int(1 + sinx )^(1//2) dx = overset(pi//2)underset0int(sin^2(x)/(2) +cos^2(x)/(2) +2sin (x)/(2) cos (x)/(2))^(1//2) dx = overset(pi//2)underset0int(sin (x)/(2) + cos (x)/(2)) dx = [-2 cos (x)/(2) + 2 sin(x)/(2)]_0^(pi//2) = 2[ sin(x)/(2) - cos (x)/(2)]_0^(pi//2)` `=2 [(sin pi//4 - cos pi//4) - (sin 0 -cos 0)] = 2[((1)/(sqrt2) - (1)/(sqrt2)) - (0-1)] =2` (iv) `overset(pi//4)underset0int (1 -cos 2x)^(1//2) dx overset(pi//4)underset0int (2sin^2 x)^(1//2) dx = sqrt2 overset(pi//4)underset0int sin x dx = sqrt2 (-cos x)_0^(pi//4)` ` = - sqrt2 [ cos pi//4 - cos 0]= - sqrt2[(1)/(sqrt2) -1]` `=-1 +sqrt2 =0.414`