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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
$\text{I}=\int\frac{(\text{x+a})^3}{\text{x}^3}\text{dx}$ is equal to:
  • ✓
    $\text{x}+3\text{a}\log\text{x}-\frac{3\text{a}^2}{\text{x}}-\frac{\text{a}^3}{2\text{x}^2}+\text{c}$
  • B
    $\text{x}^{2}+3\text{a}\log\text{x}-\frac{3\text{a}^2}{\text{x}}-\frac{\text{a}^3}{2\text{x}^2}+\text{c}$
  • C
    $\text{x}^{3}+3\text{a}\log\text{x}-\frac{2\text{a}^2}{\text{x}}-\frac{\text{3a}^3}{2\text{x}^2}+\text{c}$
  • D
    ${1}+2\text{a}\log\text{x}-\frac{2\text{a}^2}{\text{x}}-\frac{\text{3a}^3}{2\text{x}^2}+\text{c}$
✓

Answer

Correct option: A.
$\text{x}+3\text{a}\log\text{x}-\frac{3\text{a}^2}{\text{x}}-\frac{\text{a}^3}{2\text{x}^2}+\text{c}$
$\text{I}=\int\frac{(\text{x+a})^3}{\text{x}^3}\text{dx}$
$=\int\frac{\text{x}^3+\text{a}^3+3\text{ax}^2+3\text{a}^2\text{x}}{\text{x}^3}\text{dx}$
$=\int\Big(1+\frac{\text{a}^3}{\text{x}^3}+\frac{3\text{a}}{\text{x}}+\frac{3\text{a}^2}{\text{x}^2}\Big)\text{dx}$
$=\text{x}-\frac{\text{a}^3}{2\text{x}^2}+3\text{a}\log\text{x}-\frac{3\text{a}^2}{\text{x}}+\text{c}$
$\text{x}+3\text{a}\log\text{x}-\frac{3\text{a}^2}{\text{x}}-\frac{\text{a}^3}{2\text{x}^2}+\text{c}$

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