Q109 / 160 · AP EAMCET 2024 (20 May Shift 1) · Mathematics

$\int \frac{2 x^2-3}{\left(x^2-4\right)\left(x^2+1\right)} d x=\mathrm{Atan}^{-1} x+\mathrm{B} \log (x-2)+\mathrm{C} \log (x+2)$ then $6 A+7 B-5 C=$

  • Correct answer
    A
    $9$
  • B
    $10$
  • C
    $6$
  • D
    $8$

Solution

$\int \frac{\left(2 x^2-3\right)}{\left(x^2-4\right)\left(x^2+1\right)} d x=\int \frac{\left(2\left(x^2+1\right)-5\right)}{\left(x^2-4\right)\left(x^2+1\right)} d x$ $=\int \frac{2}{x^2-4} d x-\int\left(\frac{1}{x^2-4}-\frac{1}{x^2+1}\right) d x$ $=\int \frac{1}{x^2-4} d x+\int \frac{1}{x^2+1} d x=\frac{1}{4} \ln \left(\frac{x-2}{x+2}\right)+\tan ^{-1} x$ $=\frac{1}{4} \log (x-2)-\frac{1}{4} \ln (x+2)+\tan ^{-1} x$ $A=1 ; B=\frac{1}{4} ; C=-\frac{1}{4} \Rightarrow 6 A+7 B-5 C=6+\frac{7}{4}+\frac{5}{4}=9$

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

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