Let `I (a) =int_(0)^(pi) ((x)/(a)+ a sin x)^(2) dx` where `a` is positive real.
The value of `a` for which `I(a)` attains its minimum value is

Options

A. (a) `sqrt(pisqrt((2)/(3)))`
B. (b) `sqrt(pisqrt((3)/(2)))`
C. (c) `sqrt((pi)/(16))`
D. (d) `sqrt((pi)/(13))`

Answer

To solve the problem, we need to find the value of \( a \) for which the integral

\[ I(a) = \int_0^{\pi} \left( \frac{x}{a} + a \sin x \right)^2 dx \]

attains its minimum value. Let's break this down step by step.

### Step 1: Expand the integral
We start by expanding the integrand:

\[ I(a) = \int_0^{\pi} \left( \frac{x^2}{a^2} + 2 \frac{x}{a} (a \sin x) + (a \sin x)^2 \right) dx \]

This simplifies to:

\[ I(a) = \int_0^{\pi} \left( \frac{x^2}{a^2} + 2x \sin x + a^2 \sin^2 x \right) dx \]

### Step 2: Split the integral
We can split the integral into three separate integrals:

\[ I(a) = \frac{1}{a^2} \int_0^{\pi} x^2 dx + 2 \int_0^{\pi} x \sin x \, dx + a^2 \int_0^{\pi} \sin^2 x \, dx \]

### Step 3: Evaluate the integrals
1. **First integral**:

\[ \int_0^{\pi} x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^{\pi} = \frac{\pi^3}{3} \]

2. **Second integral**: We use integration by parts for \( \int_0^{\pi} x \sin x \, dx \):

Let \( u = x \) and \( dv = \sin x \, dx \). Then \( du = dx \) and \( v = -\cos x \).

Using integration by parts:

\[ \int_0^{\pi} x \sin x \, dx = \left[ -x \cos x \right]_0^{\pi} + \int_0^{\pi} \cos x \, dx = 0 + \left[ \sin x \right]_0^{\pi} = 0 \]

So,

\[ \int_0^{\pi} x \sin x \, dx = 0 \]

3. **Third integral**:

Using the identity \( \sin^2 x = \frac{1 - \cos 2x}{2} \):

\[ \int_0^{\pi} \sin^2 x \, dx = \int_0^{\pi} \frac{1 - \cos 2x}{2} \, dx = \frac{\pi}{2} \]

### Step 4: Substitute back into \( I(a) \)
Now substituting back into \( I(a) \):

\[ I(a) = \frac{1}{a^2} \cdot \frac{\pi^3}{3} + 0 + a^2 \cdot \frac{\pi}{2} \]

This simplifies to:

\[ I(a) = \frac{\pi^3}{3a^2} + \frac{\pi a^2}{2} \]

### Step 5: Find the minimum value of \( I(a) \)
To find the minimum, we differentiate \( I(a) \) with respect to \( a \) and set it to zero:

\[ \frac{dI}{da} = -\frac{2\pi^3}{3a^3} + \pi a = 0 \]

Multiplying through by \( 3a^3 \):

\[ -2\pi^3 + 3\pi a^4 = 0 \]

This gives:

\[ 3a^4 = 2\pi^3 \implies a^4 = \frac{2\pi^3}{3} \implies a = \left( \frac{2\pi^3}{3} \right)^{1/4} \]

### Step 6: Simplify \( a \)
To express \( a \) in a simpler form:

\[ a = \sqrt{\frac{2\pi}{3}} \cdot \sqrt[4]{\pi} \]

Thus, we find:

\[ a = \sqrt{\frac{2\pi}{3}} \cdot \sqrt{\pi} = \sqrt{\frac{2\pi^2}{3}} = \frac{\sqrt{2} \cdot \pi^{1/2}}{\sqrt{3}} \]

The final answer is:

\[ a = \sqrt{\frac{2\pi}{3}} \]

### Conclusion
The value of \( a \) for which \( I(a) \) attains its minimum value is:

\[ \sqrt{\frac{2\pi}{3}} \]