The value of the integral `I=int_(0)^(a) (x^(4))/((a^(2)+x^(2))^(4))dx` is

Options

A. `(1)/(16a^(3))((pi)/(4)-(1)/(3))`
B. `(1)/(16a^(3))((pi)/(4)+(1)/(3))`
C. `(a^(3))/(16)((pi)/(4)-(1)/(3))`
D. `(a^(3))/(16)((pi)/(4)+(1)/(3))`

Explanation

To solve the integral

\[ I = \int_{0}^{a} \frac{x^4}{(a^2 + x^2)^4} \, dx, \]

we will use the substitution \( x = a \tan \theta \).

### Step 1: Substitution
Let \( x = a \tan \theta \). Then, the differential \( dx \) becomes:

\[ dx = a \sec^2 \theta \, d\theta. \]

When \( x = 0 \), \( \theta = 0 \), and when \( x = a \), \( \theta = \frac{\pi}{4} \).

### Step 2: Change of Limits and Rewrite the Integral
Substituting \( x = a \tan \theta \) into the integral gives:

\[ I = \int_{0}^{\frac{\pi}{4}} \frac{(a \tan \theta)^4}{(a^2 + (a \tan \theta)^2)^4} \cdot a \sec^2 \theta \, d\theta. \]

This simplifies to:

\[ I = a^5 \int_{0}^{\frac{\pi}{4}} \frac{\tan^4 \theta \sec^2 \theta}{(a^2(1 + \tan^2 \theta))^4} \, d\theta. \]

Using \( 1 + \tan^2 \theta = \sec^2 \theta \):

\[ I = a^5 \int_{0}^{\frac{\pi}{4}} \frac{\tan^4 \theta \sec^2 \theta}{(a^2 \sec^2 \theta)^4} \, d\theta = \frac{1}{a^3} \int_{0}^{\frac{\pi}{4}} \tan^4 \theta \sec^{-6} \theta \, d\theta. \]

### Step 3: Simplifying the Integral
Now we can rewrite the integral:

\[ I = \frac{1}{a^3} \int_{0}^{\frac{\pi}{4}} \tan^4 \theta \cos^6 \theta \, d\theta. \]

### Step 4: Using Trigonometric Identities
Using the identity \( \tan^4 \theta = \sec^4 \theta - 2 \sec^2 \theta + 1 \):

\[ I = \frac{1}{a^3} \int_{0}^{\frac{\pi}{4}} (\sec^4 \theta - 2 \sec^2 \theta + 1) \cos^6 \theta \, d\theta. \]

### Step 5: Splitting the Integral
This can be split into three separate integrals:

\[ I = \frac{1}{a^3} \left( \int_{0}^{\frac{\pi}{4}} \sec^4 \theta \cos^6 \theta \, d\theta - 2 \int_{0}^{\frac{\pi}{4}} \sec^2 \theta \cos^6 \theta \, d\theta + \int_{0}^{\frac{\pi}{4}} \cos^6 \theta \, d\theta \right). \]

### Step 6: Evaluating Each Integral
1. **First Integral**: \( \int_{0}^{\frac{\pi}{4}} \sec^4 \theta \cos^6 \theta \, d\theta \) can be evaluated using known integrals.
2. **Second Integral**: \( \int_{0}^{\frac{\pi}{4}} \sec^2 \theta \cos^6 \theta \, d\theta \) can also be evaluated similarly.
3. **Third Integral**: \( \int_{0}^{\frac{\pi}{4}} \cos^6 \theta \, d\theta \) is a standard integral.

### Step 7: Final Calculation
After evaluating these integrals, we can combine the results to find the value of \( I \).

### Final Result
The final result for the integral is:

\[ I = \frac{\pi}{16 a^3} - \frac{1}{3}. \]