What is the definite integral of y=sin2x/square root(1+sin^4 x)?the upper limit of integration is pi/2 and the lower limit is 0

We'll solve the integral using
substitution.

Let (sin x)^2 =
t.

If we'll differentiate both sides, we'll
get:

2 sin x*cos x dx = dt

But
2 sin x* cos x = sin 2x => sin 2x dx = dt

We'll
re-write the integral in t:

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

Int dt/sqrt(1 + t^2) = ln [t +
sqrt(1+t^2)]

We'll apply Leibniz Newton
formula.

Since the variable x was changed, we'll change the
value of limits of integration, too.

If x = 0 =>
(sin 0)^2 = 0 = t

If x = pi/2 => (sin pi/2)^2 = 1 =
t

Int dt/sqrt(1 + t^2) = F(1) -
F(0)

F(0) = ln [0 + sqrt(1+0^2)] = ln 1 =
0

F(1) = ln [1 +
sqrt(1+1^2)]

F(1) = ln
(1+sqrt2)

F(1) - F(0) = ln
(1+sqrt2)

The requested definite integral, if
x = 0 to x = pi/2, is: Int sin 2x dx/sqrt[1+(sin x)^4] = ln
(1+sqrt2).