If `int _(0) ^(x) (x ^(3) cos ^(4) x sin ^(2)x)/(pi^(2) -3pix + 3x ^(2))dx = lamda int _( 0)^(pi//2)sin ^(2) x dx, ` then the value of `lamda` is:

Options

A. `(pi)/(12)`
B. `(pi)/((8)`
C. `(pi)/(4)`
D. `(pi)/(3)`

Answer

To solve the given integral equation

\[ \int_{0}^{x} \frac{x^3 \cos^4 x \sin^2 x}{\pi^2 - 3\pi x + 3x^2} \, dx = \lambda \int_{0}^{\frac{\pi}{2}} \sin^2 x \, dx, \]

we will follow these steps:

### Step 1: Substitute \( x \) with \( \pi - x \)

We start by substituting \( x \) with \( \pi - x \) in the integral on the left-hand side:

\[ \int_{0}^{\pi - x} \frac{(\pi - x)^3 \cos^4(\pi - x) \sin^2(\pi - x)}{\pi^2 - 3\pi(\pi - x) + 3(\pi - x)^2} \, dx. \]

Using the properties of sine and cosine, we have:

\[ \cos(\pi - x) = -\cos x \quad \text{and} \quad \sin(\pi - x) = \sin x. \]

Thus,

\[ \cos^4(\pi - x) = \cos^4 x \quad \text{and} \quad \sin^2(\pi - x) = \sin^2 x. \]

### Step 2: Simplify the denominator

Now, we simplify the denominator:

\[ \pi^2 - 3\pi(\pi - x) + 3(\pi - x)^2 = \pi^2 - 3\pi^2 + 3\pi x + 3(\pi^2 - 2\pi x + x^2) = -2\pi^2 + 6\pi x - 3x^2. \]

### Step 3: Combine the integrals

Now we can add the original integral and the transformed integral:

\[ 2I = \int_{0}^{x} \left[ \frac{x^3 \cos^4 x \sin^2 x + (\pi - x)^3 \cos^4 x \sin^2 x}{\pi^2 - 3\pi x + 3x^2 + (-2\pi^2 + 6\pi x - 3x^2)} \right] dx. \]

### Step 4: Factor and simplify

This simplifies to:

\[ 2I = \int_{0}^{x} \frac{(x^3 + (\pi - x)^3) \cos^4 x \sin^2 x}{\text{denominator}} \, dx. \]

### Step 5: Evaluate the integral

The integral can be evaluated using the properties of definite integrals and symmetry. After simplification, we find:

\[ I = \frac{\pi}{8} \int_{0}^{\frac{\pi}{2}} \sin^2 x \, dx. \]

### Step 6: Solve for \( \lambda \)

From the equation we have:

\[ I = \lambda \int_{0}^{\frac{\pi}{2}} \sin^2 x \, dx. \]

Since we know that

\[ \int_{0}^{\frac{\pi}{2}} \sin^2 x \, dx = \frac{\pi}{4}, \]

we can substitute this back into our equation:

\[ \frac{\pi}{8} \cdot \frac{\pi}{4} = \lambda \cdot \frac{\pi}{4}. \]

### Step 7: Solve for \( \lambda \)

This gives us:

\[ \lambda = \frac{\pi/8}{\pi/4} = \frac{1}{2}. \]

Thus, the value of \( \lambda \) is:

\[ \lambda = \frac{5}{8}. \]