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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
$\int \frac{e^x}{x+1}[1+(x+1) \log (x+1)] d x$ equals
  • A
    $\frac{e^x}{x+1}+c$
  • B
    $e^x \frac{x}{x+1}+c$
  • C
    $e^x \log (x+1)+e^x+c$
  • ✓
    $e^x \log (x+1)+c$
✓

Answer

Correct option: D.
$e^x \log (x+1)+c$
 Let $I=\int \frac{e^x}{x+1}[1+(x+1) \log (x+1)] d x$
$=\int e^x\left[\frac{1}{x+1}+\log (x+1)\right] d x$
It is of the form $\int e^x\left[f(x)+f^{\prime}(x) d x\right]$,
where $f(x)=\log (x+1)$ and $f^{\prime}(x)=\frac{1}{x+1}$
So, $I=e^x \log (x+1)+C$

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