Which of the following is wrong?

Options

A. `int sqrt(x^2-a^2)dx=x/2sqrt(x^2-a^2)+a^2/2 log|x+sqrt(x^2+a^2)|+c`
B. `int sqrt(x^2+a^2)dx=x/2sqrt(x^2+a^2)+a^2/2 log|x+sqrt(x^2+a^2)|+c`
C. `int 1/sqrt(x^2+a^2) dx=log|x+sqrt(x^2+a^2)|+c`
D. `int 1/sqrt(x^2-a^2) dx=log|x+sqrt(x^2-a^2)|+c`

Answer

To determine which of the given options is wrong, we will analyze the integration formulas mentioned in the transcript step by step.

### Step-by-Step Solution:

1. **Identify the Formulas**: We have several integration formulas to consider. The relevant ones from the transcript are:
- \( \int \sqrt{x^2 - a^2} \, dx = \frac{1}{2} x \sqrt{x^2 - a^2} - \frac{a^2}{2} \log(x + \sqrt{x^2 - a^2}) + C \)
- \( \int \sqrt{x^2 + a^2} \, dx = \frac{1}{2} x \sqrt{x^2 + a^2} + \frac{a^2}{2} \log(x + \sqrt{x^2 + a^2}) + C \)
- \( \int \frac{1}{\sqrt{x^2 + a^2}} \, dx = \log(x + \sqrt{x^2 + a^2}) + C \)
- \( \int \frac{1}{\sqrt{x^2 - a^2}} \, dx = \log(x + \sqrt{x^2 - a^2}) + C \)
- \( \int \frac{1}{a \sqrt{a^2 - x^2}} \, dx = \sin^{-1}\left(\frac{x}{a}\right) + C \)
- \( \int \sqrt{a^2 - x^2} \, dx = \frac{1}{2} x \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{x}{a}\right) + C \)

2. **Evaluate Each Option**:
- **Option 1**: \( \int (x^2 - a^2) \, dx = \frac{x^3}{3} - a^2 x + C \)
- This is the correct integral for \( x^2 - a^2 \).
- **Option 2**: \( \int (x^2 + a^2) \, dx = \frac{x^3}{3} + a^2 x + C \)
- This is also correct.
- **Option 3**: \( \int \frac{1}{\sqrt{x^2 + a^2}} \, dx = \log(x + \sqrt{x^2 + a^2}) + C \)
- This is correct as per the formula.
- **Option 4**: \( \int \frac{1}{\sqrt{x^2 - a^2}} \, dx = \log(x + \sqrt{x^2 - a^2}) + C \)
- This is also correct according to the formula.

3. **Identify the Wrong Option**:
- The first option is the only one that does not match the expected integration result, as it incorrectly states the integral of \( x^2 - a^2 \).

### Conclusion:
The wrong option is **Option 1**.