What is `int_0^(pi//2) (sin^3 x)/(sin^3x+cos^3x) dx`

Options

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

Answer

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 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 property of definite integrals
We can use the property:
\[ \int_0^{a} f(x) \, dx = \int_0^{a} f(a - x) \, dx \]
In our case, let \( a = \frac{\pi}{2} \). Thus, 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 identity \( \sin\left(\frac{\pi}{2} - x\right) = \cos x \) 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 equations 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:
\[ \int_0^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_0^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2} \]

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

### Final Answer
The value of the integral is:
\[ \int_0^{\frac{\pi}{2}} \frac{\sin^3 x}{\sin^3 x + \cos^3 x} \, dx = \frac{\pi}{4} \]