What is the definite integral of ysin2xsquare root1sin4 x th

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

Solution

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).