Question

Evaluate: $\int \frac{dx}{(2x-7)\sqrt{(x-3)(x-4)}}$.

Answer 1

$\int \frac{dx}{(2x-7)\sqrt{(x-3)(x-4)}}$

$\int \frac{dx}{(2x-7)\sqrt{x^2-7x+12}}$

$\int \frac{dx}{2(2x-7)\sqrt{x^2-7x+12}}$

$\int \frac{2dx}{(2x-7)\sqrt{4x^2-28x+48}}$

$\int \frac{2dx}{(2x-7)\sqrt{4x^2-28x+49-1}}$

$\int \frac{2dx}{(2x-7)\sqrt{(2x-7{)}^2-1}}$

Let $2x-7=t$

$2dx=dt$

$\int \frac{dt}{t\sqrt{t^2-1}}$

$={\sec }^{-1}\left(t\right)+c$

$={\sec }^{-1}(2x-7)+c$.