The value of `int (dx)/(sqrt (a-x))`

Options

A. `2 sqrt ( a- x ) + c`
B. `-2 sqrt (a- x ) + c`
C. `sqrtx + c`
D. `x +c`

Answer

To solve the integral \( I = \int \frac{dx}{\sqrt{a - x}} \), we will use a substitution method. Here’s a step-by-step solution:

### Step 1: Set up the integral
We start with the integral:
\[ I = \int \frac{dx}{\sqrt{a - x}} \]

### Step 2: Use substitution
Let’s make the substitution:
\[ t = \sqrt{a - x} \]
Then, squaring both sides gives:
\[ t^2 = a - x \implies x = a - t^2 \]

### Step 3: Differentiate to find \( dx \)
Now, we differentiate \( x \) with respect to \( t \):
\[ dx = -2t \, dt \]

### Step 4: Substitute \( dx \) and \( t \) into the integral
Substituting \( dx \) and \( \sqrt{a - x} \) into the integral gives:
\[ I = \int \frac{-2t \, dt}{t} = \int -2 \, dt \]

### Step 5: Integrate
Now, we can integrate:
\[ I = -2t + C \]

### Step 6: Substitute back for \( t \)
Now, we substitute back \( t = \sqrt{a - x} \):
\[ I = -2\sqrt{a - x} + C \]

### Final Result
Thus, the value of the integral is:
\[ I = -2\sqrt{a - x} + C \]

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