Q116 / 160 · AP EAMCET 2023 (16 May Shift 2) · Mathematics

If $\int_3^b \frac{x-1}{2 x-x^2} d x=\frac{1}{2}$, then $(b-1)^2=$

  • A
    $2$
  • B
    $\sqrt{2}$
  • Correct answer
    C
    $1+\frac{3}{\mathrm{e}}$
  • D
    $\sqrt{\frac{3}{\mathrm{e}}-1}$

Solution

Given $\int_3^b \frac{x-1}{2 x-x^2} d x=\frac{1}{2} \Rightarrow-\frac{1}{2} \int_3^b \frac{-2 x+2}{2 x-x^2} d x=\frac{1}{2}$ $\begin{aligned} & \Rightarrow\left(\ln \left|2 x-x^2\right|\right)_3^b=-1 \Rightarrow \ln \left|2 b-b^2\right|-\ln |6-9|=-1 \\ & \Rightarrow \ln \left|2 b-b^2\right|=\ln 3-1 \Rightarrow \ln \left|b^2-2 b\right|=\ln (3)-1 \\ & \Rightarrow(b-1)^2=\frac{3}{e}+1\end{aligned}$

Hence, the correct answer is option: 'C'.

More Definite Integration Questions