If `|(e^x,sinx),(cosx,log_e(1+x^2))|=p+qx+rx^2+.....` then p =...... and q = …….

Answer

To solve the determinant \( |(e^x, \sin x), (\cos x, \log_e(1+x^2))| \) and express it in the form \( p + qx + rx^2 + \ldots \), we will follow these steps:

### Step 1: Calculate the Determinant
The determinant of a 2x2 matrix \( |(a, b), (c, d)| \) is given by the formula:
\[ ad - bc \]
For our case, we have:
- \( a = e^x \)
- \( b = \sin x \)
- \( c = \cos x \)
- \( d = \log_e(1+x^2) \)

Thus, the determinant becomes:
\[ D = e^x \cdot \log_e(1+x^2) - \sin x \cdot \cos x \]

### Step 2: Expand the Determinant
We need to expand \( D \) around \( x = 0 \) to find the coefficients \( p \) and \( q \). We will use Taylor series expansions for \( e^x \), \( \sin x \), \( \cos x \), and \( \log_e(1+x^2) \).

1. **Taylor Series Expansions**:
- \( e^x = 1 + x + \frac{x^2}{2} + O(x^3) \)
- \( \sin x = x - \frac{x^3}{6} + O(x^5) \)
- \( \cos x = 1 - \frac{x^2}{2} + O(x^4) \)
- \( \log_e(1+x^2) = x^2 - \frac{x^4}{4} + O(x^6) \)

### Step 3: Substitute the Expansions
Substituting these expansions into the determinant \( D \):
\[ D = (1 + x + \frac{x^2}{2}) \cdot (x^2 - \frac{x^4}{4}) - (x - \frac{x^3}{6})(1 - \frac{x^2}{2}) \]

### Step 4: Simplify the Expression
Expanding the first term:
\[ (1 + x + \frac{x^2}{2})(x^2 - \frac{x^4}{4}) = x^2 + x^3 + \frac{x^4}{2} - \frac{x^4}{4} - \frac{x^5}{4} - \frac{x^6}{8} \]
Combining like terms gives:
\[ x^2 + x^3 + \frac{x^4}{4} - \frac{x^5}{4} + O(x^6) \]

Now expanding the second term:
\[ (x - \frac{x^3}{6})(1 - \frac{x^2}{2}) = x - \frac{x^3}{6} - \frac{x^3}{2} + O(x^5) = x - \left(\frac{1}{6} + \frac{1}{2}\right)x^3 + O(x^5) = x - \frac{2}{3}x^3 + O(x^5) \]

### Step 5: Combine Both Parts
Now, we combine both parts:
\[ D = (x^2 + x^3 + \frac{x^4}{4} - \frac{x^5}{4}) - (x - \frac{2}{3}x^3) \]
This simplifies to:
\[ D = -x + x^2 + \left(1 + \frac{2}{3}\right)x^3 + \frac{x^4}{4} - \frac{x^5}{4} + O(x^6) \]
Thus, we have:
\[ D = -x + x^2 + \frac{5}{3}x^3 + \frac{x^4}{4} + O(x^5) \]

### Step 6: Identify Coefficients
From the expression \( D = p + qx + rx^2 + \ldots \):
- The constant term \( p = 0 \)
- The coefficient of \( x \) is \( q = -1 \)

### Final Answers
Thus, we have:
- \( p = 0 \)
- \( q = -1 \)