If ∫(2x - 5)/((2x - 3)^3)e^2x dx = (λ e^2x)/((2x - 3)^2) + C , where…

The correct answer is Option (1) → \(\displaystyle\frac{1}{2}\)

Since integration and differentiation are inverse processes, the derivative of the right side must equal the integrand. By the product rule,

    \( \displaystyle \frac{d}{dx}\left[e^{2x}(2x - 3)^{-2}\right] = 2e^{2x}(2x - 3)^{-2} - 4e^{2x}(2x - 3)^{-3} \)

    \( \displaystyle = \frac{e^{2x}\left[2(2x - 3) - 4\right]}{(2x - 3)^3} \)

    \( \displaystyle = \frac{2(2x - 5)}{(2x - 3)^3}e^{2x} \)

So

    \( \displaystyle \frac{d}{dx}\left[\frac{\lambda e^{2x}}{(2x - 3)^2}\right] = 2\lambda \cdot \frac{2x - 5}{(2x - 3)^3}e^{2x} \)

This equals the integrand when

    \( \displaystyle 2\lambda = 1,\quad \lambda = \frac{1}{2} \)