Evaluate the following definite integrals:
`int_(0)^(pi//2)(sin 2x)/(sin^(4)x+cos^(4)x)dx`

Answer

To evaluate the definite integral

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin 2x}{\sin^4 x + \cos^4 x} \, dx, \]

we will follow these steps:

### Step 1: Simplify the Denominator
We start by simplifying the expression \(\sin^4 x + \cos^4 x\). We can use the identity:

\[ a^2 + b^2 = (a + b)^2 - 2ab. \]

Let \(a = \sin^2 x\) and \(b = \cos^2 x\). Then,

\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x. \]

Since \(\sin^2 x + \cos^2 x = 1\), we have:

\[ \sin^4 x + \cos^4 x = 1 - 2\sin^2 x \cos^2 x. \]

Using the identity \(\sin 2x = 2\sin x \cos x\), we can express \(\sin^2 x \cos^2 x\) as:

\[ \sin^2 x \cos^2 x = \frac{\sin^2 2x}{4}. \]

Thus, we can rewrite the denominator:

\[ \sin^4 x + \cos^4 x = 1 - \frac{1}{2}\sin^2 2x. \]

### Step 2: Substitute the Denominator into the Integral
Now substituting this back into the integral, we get:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin 2x}{1 - \frac{1}{2}\sin^2 2x} \, dx. \]

### Step 3: Factor Out Constants
We can factor out \(\frac{1}{2}\) from the denominator:

\[ I = 2 \int_{0}^{\frac{\pi}{2}} \frac{\sin 2x}{2 - \sin^2 2x} \, dx. \]

### Step 4: Use Substitution
Let \(t = \cos 2x\). Then, we have:

\[ dt = -2\sin 2x \, dx \quad \Rightarrow \quad \sin 2x \, dx = -\frac{dt}{2}. \]

The limits change as follows: when \(x = 0\), \(t = \cos(0) = 1\) and when \(x = \frac{\pi}{2}\), \(t = \cos(\pi) = -1\). Thus, we can rewrite the integral:

\[ I = 2 \int_{1}^{-1} \frac{-\frac{dt}{2}}{2 - (1 - t^2)} = 2 \int_{-1}^{1} \frac{dt}{1 + t^2}. \]

### Step 5: Evaluate the Integral
The integral \(\int \frac{dt}{1 + t^2}\) is known to be \(\tan^{-1}(t)\). Thus, we have:

\[ I = 2 \left[ \tan^{-1}(t) \right]_{-1}^{1} = 2 \left( \tan^{-1}(1) - \tan^{-1}(-1) \right). \]

Since \(\tan^{-1}(1) = \frac{\pi}{4}\) and \(\tan^{-1}(-1) = -\frac{\pi}{4}\), we find:

\[ I = 2 \left( \frac{\pi}{4} - (-\frac{\pi}{4}) \right) = 2 \left( \frac{\pi}{4} + \frac{\pi}{4} \right) = 2 \cdot \frac{\pi}{2} = \pi. \]

### Final Answer
Thus, the value of the definite integral is:

\[ \boxed{\pi}. \]