Q63 / 100 · BITSAT 2024 (Memory Based Paper 2) · Mathematics

$\int_5^9 \frac{\log 3 x^2}{\log 3 x^2+\log \left(588-84 x+3 x^2\right)} d x$ is equal to

  • Correct answer
    A
    $2$
  • B
    $1$
  • C
    $\frac{1}{2}$
  • D
    $4$

Solution

Let $I=\int_5^9 \frac{\log 3 x^2 d x}{\log 3 x^2+\log \left(588-84 x+3 x^2\right)}$ ...(i) $\begin{aligned} & =\int_5^9 \frac{\log 3 x^2 d x}{\log 3 x^2+\log 3(14-x)^2} \\ & =\int_5^9 \frac{\log 3(14-x)^2 d x}{\log 3(14-x)^2+\log 3(14-(14-x))^2} \\ & \quad\left[\int_a^b f(x) d x=\int_a^b f(a+b-x) d x\right]\end{aligned}$ $I=\int_5^9 \frac{\log 3(14-x)^2 d x}{\log 3(14-x)^2+\log 3 x^2}$ ...(ii) Adding Eqs. (i) and (ii), we get $\begin{aligned} & 2 I=\int_5^9 \frac{\log 3 x^2+\log 3(14-x)^2}{\log 3(14-x)^2+\log 3 x^2} d x \\ & 2 \mathrm{I}=\int_5^9 d x=9-5=4 \Rightarrow I=2 \end{aligned}$

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

More Definite Integration Questions