Question

The integrating factor of \( y + \frac{d}{dx}(xy) = x(\sin x + \log x) \) is

• A. \( x \)
• B. \( \log x^2 \)
• C. \( x^2 \)
• D. \( x^3 \)

Hint:

$e^{\log(f(x))} = f(x)$. Always simplify the exponent in I.F.

Answer 1

The Correct Option is A

Step 1: Expand Derivative
$y + [x \frac{dy}{dx} + y] = x(\sin x + \log x)$. $x \frac{dy}{dx} + 2y = x(\sin x + \log x)$.
Step 2: Standard Form
Divide by $x$: $\frac{dy}{dx} + \frac{2}{x}y = \sin x + \log x$. Here $P = 2/x$.
Step 3: Calculate I.F.
$I.F. = e^{\int (2/x) dx} = e^{2 \log x} = e^{\log x^2} = x^2$.
Step 4: Conclusion
The integrating factor is $x^2$. (Correction: Option C is correct based on math).
Final Answer:(C)