Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
To evaluate $\int \frac{dx}{(x-1)\sqrt{-x^2 + 3x - 2}}$, one of the most suitable substitution could be:
$\sqrt{-x^2 + 3x - 2} = u$
$\sqrt{-x^2 + 3x - 2} = (u\sqrt{2})$
$\sqrt{-x^2 + 3x - 2} = u(1 - u)$
$\sqrt{-x^2 + 3x - 2} = u(x - 2)$
Step-by-Step Solution
Key Concept: Factoring the quadratic and choosing an appropriate root form based on the sign conditions determines the correct substitution.
With $m = -1 < 0$ and $P = -2 < 0$, we factorize $-x^2 + 3x - 2 = -(x-1)(x-2)$. Using case 3, we write $\sqrt{-x^2+3x-2} = u(x-2)$ or $(x-1)u$ or $u(1-x)$ depending on the domain selection.
Correct Answer: 4