The general solution of the differential equation \(\displaystyle\frac{dy}{dx} = \frac{x}{1 + x^2}\) is (C is an arbitrary constant)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(y = \displaystyle\frac{1}{2}\log(1 + x^2) + C\)
The right side contains only x, so
\( \displaystyle y = \int \frac{x}{1 + x^2}\,dx \)
The numerator is half the derivative of the denominator, since \(\frac{d}{dx}(1 + x^2) = 2x\):
\( \displaystyle y = \frac{1}{2}\int \frac{2x}{1 + x^2}\,dx \)
\( \displaystyle y = \frac{1}{2}\log(1 + x^2) + C \)
The answer \(\tan^{-1}x + C\) belongs to \(\frac{1}{1 + x^2}\), where the numerator has no x.