`int_(0)^(pi//2) (sqrt""(sin^(3).x) dx)/(sqrt""(sin^(3) x) + sqrt""(cos^(3) x))`=

Options

A. `pi//4`
B. 1
C. 0
D. `pi//2`

Answer

To solve the integral

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin^3 x}}{\sqrt{\sin^3 x} + \sqrt{\cos^3 x}} \, dx, \]

we will use a property of definite integrals.

### Step 1: Define the integral
Let

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin^3 x}}{\sqrt{\sin^3 x} + \sqrt{\cos^3 x}} \, dx. \]

### Step 2: Use the property of definite integrals
We can use the property

\[ \int_{0}^{A} f(x) \, dx = \int_{0}^{A} f(A - x) \, dx. \]

For our case, we set \( A = \frac{\pi}{2} \):

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin^3\left(\frac{\pi}{2} - x\right)}}{\sqrt{\sin^3\left(\frac{\pi}{2} - x\right)} + \sqrt{\cos^3\left(\frac{\pi}{2} - x\right)}} \, dx. \]

### Step 3: Simplify the expression
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:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\cos^3 x}}{\sqrt{\cos^3 x} + \sqrt{\sin^3 x}} \, dx. \]

### Step 4: Combine the two integrals
Now we have two expressions for \( I \):

1. \( I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin^3 x}}{\sqrt{\sin^3 x} + \sqrt{\cos^3 x}} \, dx \)
2. \( I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\cos^3 x}}{\sqrt{\cos^3 x} + \sqrt{\sin^3 x}} \, dx \)

Adding these two integrals:

\[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{\sqrt{\sin^3 x}}{\sqrt{\sin^3 x} + \sqrt{\cos^3 x}} + \frac{\sqrt{\cos^3 x}}{\sqrt{\cos^3 x} + \sqrt{\sin^3 x}} \right) dx. \]

### Step 5: Simplify the sum
The sum simplifies to:

\[ 2I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin^3 x} + \sqrt{\cos^3 x}}{\sqrt{\sin^3 x} + \sqrt{\cos^3 x}} \, dx = \int_{0}^{\frac{\pi}{2}} 1 \, dx. \]

### 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 gives:

\[ I = \frac{\pi}{4}. \]

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

\[ \boxed{\frac{\pi}{4}}. \]

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