`l_(n)=intx^(n)sqrt(a^(2)-x^(2))dx`, then answer the following questions:
The value of `l_(1)` is

Options

A. `(2)/(3)(a^(2)-x^(2))^(1//2)`
B. `(1)/(3)(a^(2)-x^(2))^(3//2)`
C. `-(2)/(3)(a^(2)-x^(2))^(3//2)`
D. `-(1)/(3)(a^(2)-x^(2))^(3//2)`

Answer

To find the value of \( I_1 = \int x \sqrt{a^2 - x^2} \, dx \), we will follow these steps:

### Step 1: Substitution
We start by substituting \( t = a^2 - x^2 \). Then, we differentiate both sides:
\[ dt = -2x \, dx \quad \Rightarrow \quad dx = -\frac{dt}{2x} \]

### Step 2: Rewrite the Integral
Now we can rewrite the integral \( I_1 \) using our substitution:
\[ I_1 = \int x \sqrt{t} \left(-\frac{dt}{2x}\right) \]
This simplifies to:
\[ I_1 = -\frac{1}{2} \int \sqrt{t} \, dt \]

### Step 3: Integrate
Now we integrate \( \sqrt{t} \):
\[ \int \sqrt{t} \, dt = \frac{t^{3/2}}{\frac{3}{2}} = \frac{2}{3} t^{3/2} \]
Thus, we have:
\[ I_1 = -\frac{1}{2} \cdot \frac{2}{3} t^{3/2} + C = -\frac{1}{3} t^{3/2} + C \]

### Step 4: Substitute Back
Now we substitute back \( t = a^2 - x^2 \):
\[ I_1 = -\frac{1}{3} (a^2 - x^2)^{3/2} + C \]

### Final Answer
The value of \( I_1 \) is:
\[ I_1 = -\frac{1}{3} (a^2 - x^2)^{3/2} + C \]

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