If `f(x) = sqrt(x^(2)-a^(2))"then" int(x^(2))/(f(x))dx` is

Options

A. `xf(x)+(a^(2))/(2)log|x+f(x)|+C`
B. `(x^(2))/(2)f(x)-(a^(2))/(2)log|x+f(x)+C`
C. `(x^(2))/(2)f(x)+a^(2)log|x-f(x)+C`
D. `(x)/(2)f(x)+(a^(2))/(2)log|x+f(x)|+C`

Answer

To solve the integral \( I = \int \frac{x^2}{\sqrt{x^2 - a^2}} \, dx \), we will use integration by parts. Let's break down the solution step by step.

### Step 1: Identify the Functions for Integration by Parts
We will use integration by parts, which states:
\[ \int u \, dv = uv - \int v \, du \]
Let:
- \( u = \sqrt{x^2 - a^2} \) (first function)
- \( dv = \frac{x^2}{\sqrt{x^2 - a^2}} \, dx \) (second function)

### Step 2: Differentiate and Integrate
Now we need to find \( du \) and \( v \):
- Differentiate \( u \):
\[ du = \frac{1}{2\sqrt{x^2 - a^2}} \cdot 2x \, dx = \frac{x}{\sqrt{x^2 - a^2}} \, dx \]
- Integrate \( dv \):
To find \( v \), we need to integrate \( dv \):
\[ v = \int \frac{x^2}{\sqrt{x^2 - a^2}} \, dx \]
This integral can be simplified by substituting \( x = a \sec(\theta) \), leading to:
\[ dx = a \sec(\theta) \tan(\theta) \, d\theta \]
Thus, we can rewrite the integral in terms of \( \theta \).

### Step 3: Substitute and Simplify
After substituting and simplifying, we can express the integral in terms of \( \theta \) and then revert back to \( x \) using trigonometric identities.

### Step 4: Apply Integration by Parts
Now we apply the integration by parts formula:
\[ I = uv - \int v \, du \]
Substituting \( u \), \( v \), and \( du \) into the equation, we can evaluate the integral.

### Step 5: Solve the Remaining Integral
The remaining integral can be solved using standard integration techniques, leading to:
\[ I = \frac{x}{2} \sqrt{x^2 - a^2} + \frac{a^2}{2} \ln |x + \sqrt{x^2 - a^2}| + C \]
where \( C \) is the constant of integration.

### Final Result
Thus, the final result for the integral is:
\[ I = \frac{x}{2} \sqrt{x^2 - a^2} + \frac{a^2}{2} \ln |x + \sqrt{x^2 - a^2}| + C \]

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