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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
Choose the correct option from given four options$:\ \int\frac{\text{x}}{\text{x}+1}$ is equal to:
  • A
    $\text{x}+\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
  • B
    $\text{x}+\frac{\text{x}^2}{2}-\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
  • C
    $\text{x}-\frac{\text{x}^2}{2}-\frac{\text{x}^3}{3}-\log|1+\text{x}|+\text{C}$
  • ✓
    $\text{x}-\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1+\text{x}|+\text{C}$
✓

Answer

Correct option: D.
$\text{x}-\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1+\text{x}|+\text{C}$
Let $\text{I}=\int\frac{\text{x}}{\text{x}+1}\text{dx}$
We know that, $\frac{\text{x}^3}{\text{x}+1}$ is an improper fraction.
To convert it into proper fraction, we have to divide numerator by denominator.
After performing long division, we get
$\frac{\text{x}^3}{\text{x}+1}=(\text{x}^2-\text{x}+1)-\frac{1}{(\text{x}+1)}$
$\therefore\ \text{I}=\int\Big((\text{x}^2-\text{x}+1)-\frac{1}{(\text{x}+1)}\Big)\text{dx}$
$=\frac{\text{x}^3}{3}-\frac{\text{x}^2}{2}+\text{x}-\log|\text{x}+1|+\text{C}$

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