`int(sin^(4)x)/(sin^(4)x+cos^(4)x)dx` is equal to

Options

A. `(1)/(2){x+(1)/(2sqrt(2))log|(sqrt(2)+sin2x)/(sqrt(2)-sin2x)|}+C`
B. `(1)/(2){x+(1)/(2sqrt(2))log|(1+sin2x)/(1-sin2x)|}+C`
C. `(1)/(2){x+(1)/(2sqrt(2))log|(sqrt(2)+sin2x)/(sqrt(2)-sin2x)|}+C`
D. `(1)/(2){x+(1)/(2sqrt(2))log|(1+sqrt(2)sin2x)/(1-sqrt(2)sin2x)|}+C`

Explanation

To solve the integral \( I = \int \frac{\sin^4 x}{\sin^4 x + \cos^4 x} \, dx \), we will follow a series of steps to simplify and evaluate the integral.

### Step 1: Multiply and Divide by 2
We start by multiplying and dividing the integrand by 2:
\[ I = \frac{1}{2} \int \frac{2 \sin^4 x}{\sin^4 x + \cos^4 x} \, dx \]

### Step 2: Add and Subtract \(\cos^4 x\)
Next, we add and subtract \(\cos^4 x\) in the numerator:
\[ I = \frac{1}{2} \int \frac{2 \sin^4 x + \cos^4 x - \cos^4 x}{\sin^4 x + \cos^4 x} \, dx \]
This can be rewritten as:
\[ I = \frac{1}{2} \int \left( \frac{\sin^4 x + \cos^4 x}{\sin^4 x + \cos^4 x} + \frac{\sin^4 x - \cos^4 x}{\sin^4 x + \cos^4 x} \right) \, dx \]
The first term simplifies to 1:
\[ I = \frac{1}{2} \int dx + \frac{1}{2} \int \frac{\sin^4 x - \cos^4 x}{\sin^4 x + \cos^4 x} \, dx \]

### Step 3: Simplify the Second Integral
Now, we can simplify the second integral:
\[ \sin^4 x - \cos^4 x = (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x) \]
Since \(\sin^2 x + \cos^2 x = 1\), we have:
\[ \sin^4 x - \cos^4 x = \sin^2 x - \cos^2 x \]
Thus, we rewrite the integral:
\[ I = \frac{1}{2} \int dx + \frac{1}{2} \int \frac{\sin^2 x - \cos^2 x}{\sin^4 x + \cos^4 x} \, dx \]

### Step 4: Substitute for \(\sin^2 x - \cos^2 x\)
We know that \(\sin^2 x - \cos^2 x = -\cos 2x\), so we can substitute:
\[ I = \frac{1}{2} \int dx - \frac{1}{2} \int \frac{\cos 2x}{\sin^4 x + \cos^4 x} \, dx \]

### Step 5: Rewrite the Denominator
Next, we rewrite the denominator:
\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x = 1 - 2\sin^2 x \cos^2 x \]
Let \(u = \sin 2x\), then \(du = 2\cos 2x \, dx\), or \(dx = \frac{du}{2\cos 2x}\).

### Step 6: Change of Variables
Substituting into the integral:
\[ I = \frac{1}{2} x - \frac{1}{4} \int \frac{du}{1 - \frac{1}{2}u^2} \]

### Step 7: Solve the Integral
The integral \(\int \frac{du}{1 - \frac{1}{2}u^2}\) can be solved using the formula for the integral of the form \(\int \frac{dx}{a^2 - x^2}\):
\[ \int \frac{du}{1 - \frac{1}{2}u^2} = \frac{2}{\sqrt{2}} \text{arctanh}\left(\frac{u}{\sqrt{2}}\right) + C \]

### Final Step: Combine Results
Putting everything together, we have:
\[ I = \frac{1}{2} x - \frac{1}{4} \cdot \frac{2}{\sqrt{2}} \text{arctanh}\left(\frac{\sin 2x}{\sqrt{2}}\right) + C \]
Thus, the final result is:
\[ I = \frac{1}{2} x - \frac{1}{2\sqrt{2}} \text{arctanh}\left(\frac{\sin 2x}{\sqrt{2}}\right) + C \]