What is `int_(0)^(pi//2) (sin^(3)x)/(sin^(3)x + cos^(3)x) dx` ?

Options

A. `pi`
B. `pi/2`
C. `pi/4`
D. `0`

Explanation

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^3 x}{\sin^3 x + \cos^3 x} \, dx \), we can use a symmetry property of definite integrals. Here’s a step-by-step solution:

### Step 1: Define the Integral
Let
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^3 x}{\sin^3 x + \cos^3 x} \, dx \]

### Step 2: Use the Symmetry Property
We can use the property of definite integrals:
\[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx \]
In our case, \( a = 0 \) and \( b = \frac{\pi}{2} \), so we have:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^3\left(\frac{\pi}{2} - x\right)}{\sin^3\left(\frac{\pi}{2} - x\right) + \cos^3\left(\frac{\pi}{2} - x\right)} \, dx \]

### Step 3: Simplify the Integral
Using the identities \( \sin\left(\frac{\pi}{2} - x\right) = \cos x \) and \( \cos\left(\frac{\pi}{2} - x\right) = \sin x \), we can rewrite the integral:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\cos^3 x}{\cos^3 x + \sin^3 x} \, dx \]

### Step 4: Combine the Two Integrals
Now we have two expressions for \( I \):
1. \( I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^3 x}{\sin^3 x + \cos^3 x} \, dx \)
2. \( I = \int_{0}^{\frac{\pi}{2}} \frac{\cos^3 x}{\cos^3 x + \sin^3 x} \, dx \)

Adding these two integrals gives:
\[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{\sin^3 x}{\sin^3 x + \cos^3 x} + \frac{\cos^3 x}{\sin^3 x + \cos^3 x} \right) \, dx \]

### Step 5: Simplify the Sum
The sum simplifies to:
\[ 2I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^3 x + \cos^3 x}{\sin^3 x + \cos^3 x} \, dx = \int_{0}^{\frac{\pi}{2}} 1 \, dx \]

### Step 6: Evaluate the Integral
Now we can evaluate the integral:
\[ 2I = \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2} \]

### Step 7: Solve for \( I \)
Dividing both sides by 2, we find:
\[ I = \frac{\pi}{4} \]

### Final Result
Thus, the value of the integral is:
\[ \int_{0}^{\frac{\pi}{2}} \frac{\sin^3 x}{\sin^3 x + \cos^3 x} \, dx = \frac{\pi}{4} \]