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Definite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
MCQ
$\int_3^8 \frac{2-3 x}{x \sqrt{(1+x)}} d x$ is equal to
  • ✓
    $2 \log \left(\frac{3}{2 e ^3}\right)$
  • B
    $\log \left(\frac{3}{e^3}\right)$
  • C
    $4 \log \left(\frac{3}{e^3}\right)$
  • D
    $\log \left(\frac{3}{2 e ^3}\right)$
✓

Answer

Correct option: A.
$2 \log \left(\frac{3}{2 e ^3}\right)$
(A)
Put $x+1= t ^2 \Rightarrow d x=2 t dt$
When $x=3, t =2$ and when $x=8, t =3$
$\therefore \quad \int_3^8 \frac{2-3 x}{x \sqrt{1+x}} d x=2 \int_2^3 \frac{2-3\left( t ^2-1\right)}{ t ^2-1} dt$
$=2 \int_2^3\left(\frac{2}{t^2-1}-3\right) d t$
$=2\left[2 \cdot \frac{1}{2 \times 1} \log \left(\frac{ t -1}{ t +1}\right)-3 t \right]_2^3$
$=2\left(\log \frac{1}{2}-\log \frac{1}{3}-3\right)$
$=2\left(\log \frac{3}{2}-3 \log e \right)=2\left(\log \frac{3}{2}-\log e ^3\right)$
$=2 \log \left(\frac{3}{2 e ^3}\right)$

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