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Indefinite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integration1 Mark
MCQ
$\int \frac{\log (3 x)}{x \log (9 x)} \cdot d x=$
  • A
    $log(3x) – log(9x) + c$
  • B
    $log(x) – (log 3) . log(log 9x) + c$
  • C
    $\log 9-(\log x) \cdot \log (\log 3 x)+c$
  • ✓
    $\log (x)+\log (3) \cdot \log (\log 9 x)+c$
✓

Answer

Correct option: D.
$\log (x)+\log (3) \cdot \log (\log 9 x)+c$
$log(x) – (log 3) . log(log 9x) + c$
Hint :
$\int \frac{\log 3 x}{x \log (9 x)} d x=\int \frac{\log \left(\frac{9 x}{3}\right)}{x \log (9 x)} d x$
$=\int \frac{\log (9 x)-\log 3}{x \log (9 x)} d x$
$=\int\left[\frac{1}{x}-\frac{\log 3}{x \log (9 x)}\right] d x$
$=\int \frac{1}{x} d x-(\log 3) \int \frac{\left(\frac{1}{x}\right)}{\log (9 x)} d x$
$=\log (x)-(\log 3) \cdot \log (\log 9 x)+c .$

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