We have to determine the definite integral of y = sin 2x /sqrt(1 + (sin x)^4), x = 0 to x = pi/2

Int [ sin 2x /sqrt(1 + (sin x)^4) dx]

let 1 + (sin x)^2 = y

dy/dx = 2*sin x* cos x = sin 2x

=> dy = sin 2x dx

Int [ sin 2x /sqrt(1 + (sin x)^4) dx]

=> Int [ 1/sqrt ( 1 + y^2) dy]

=> arcsinh(y)

substitute y = 1 + (sin x)^2

=> arcsinh(1 + (sin x)^2) + C

Between the limits x = 0 and x = pi/2, the definite integral is

arcsinh(1 + 1) - arcsinh(1)

=> arcsinh(2) - arcsinh(1)

**The definite integral is arcsinh(2) - arcsinh(1)**

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