Question

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

• A. ${\sin }^{-1}(2x-3)+c$
• B. ${\sin }^{-1}(2x+5)+C$
• C. ${\sin }^{-1}(3–2x)+C$
• D. ${\sin }^{-1}(5–2x)+C$

Answer 1

The correct option is A

${\sin }^{-1}(2x-3)+c$

Explanation for the correct option:

Evaluating the integral:

$\begin{array}{l}\begin{array}{rl}\int \frac{dx}{\sqrt{(1-x)(x-2)}}& =\int \frac{dx}{\sqrt{-x^2+3x-2}}\\ & =\int \frac{dx}{\sqrt{-(x^2-3x+2)}}\\ & =\int \frac{dx}{\sqrt{-\left[{\left(x-\frac{3}{2}\right)}^2-\frac{1}{4}\right]}}\\ & =\int \frac{dx}{\sqrt{{\left(\frac{1}{2}\right)}^2-{\left(x-\frac{3}{2}\right)}^2}}\\ & ={\sin }^{-1}\left(\frac{x-\frac{3}{2}}{\frac{1}{2}}\right)+c\\ \therefore \int \frac{dx}{\sqrt{(1-x)(x-2)}}& ={\sin }^{-1}(2x-3)+C\end{array}\end{array}$

Hence option (A) is correct.