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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
What is $\int\frac{\text{dx}}{\text{x}(1+\text{lnx})^\text{n}}$ equal to $(\text{n}\neq1)$
  • A
    $\frac{{1}}{{(\text{n}-1)}(1+\text{lnx})^{\text{n}-1}}+\text{c}$
  • B
    $\frac{1-\text{n}}{(1+\text{lnx})^{1-\text{n}}}+\text{c}$
  • C
    $\frac{{\text{n}+1}}{{(1+\text{lnx})}^{\text{n}+1}}+\text{c}$
  • ✓
    $-\frac{1}{(\text{n}+1)(1+\text{lnx})^{\text{n}-1}}+\text{c}$
✓

Answer

Correct option: D.
$-\frac{1}{(\text{n}+1)(1+\text{lnx})^{\text{n}-1}}+\text{c}$
Given,
$\int\frac{1}{\text{x}(1+\text{ln}(\text{x}))^\text{n}}\text{dx}$
apply $u = 1 + ln (x)$
$=\int\frac{1}{\text{u}^{\text{n}}}\text{du}$
$=\int\text{u}^{-\text{n}}\text{du}$
$=\frac{\text{u}^{-\text{n}+1}}{-\text{n}+1}$
$=\frac{(1+\text{ln}(\text{x}))^{-{\text{n}+1}}}{{-\text{n}+1}}$
$-\frac{1}{(\text{n}+1)(1+\text{lnx})^{\text{n}-1}}+\text{c}$

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