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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
Choose the correct answer in exercise$:\ \int\frac{\text{x dx}}{(\text{x}-1)(\text{x}-2)}$ equals
  • A
    $\text{log}\Bigg|\frac{(\text{x}-1)^2}{\text{x}-2}\Bigg|+\text{C}$
  • ✓
    $\text{log}\Bigg|\frac{(\text{x}-2)^2}{\text{x}-1}\Bigg|+\text{C}$
  • C
    $\text{log}\Bigg|\Bigg(\frac{\text{x}-1}{\text{x}-2}\Bigg)^2\Bigg|+\text{C}$
  • D
    $\text{log}|(\text{x}-1)(\text{x}-2)|+\text{c}$
✓

Answer

Correct option: B.
$\text{log}\Bigg|\frac{(\text{x}-2)^2}{\text{x}-1}\Bigg|+\text{C}$
Let, $\frac{\text{x}}{(\text{x}-1)(\text{x}-2)}=\frac{\text{A}}{\text{x}-1}+\frac{\text{B}}{\text{x}-2}\dots(\text{i})$
$\Rightarrow x = A(x - 2) + B(x -1)$
$\Rightarrow x = Ax - 2A + Bx - B$
Comparing coefficients of $x:$
$A + B = 1…….(ii)$
Comparing constants
$–2A – B = 0…….(iii)$
On solving eq. $(ii)$ and $(iii),$ we get
$A = –1, B = 2$
Putting these values of $A$ and $B$ in eq. $(i),$
$\frac{\text{x}}{(\text{x}-1)(\text{x}-2)}=\frac{-1}{\text{x}-1}+\frac{2}{\text{x}-2}$
$\Rightarrow\ \int\frac{\text{x}}{(\text{x}-1)(\text{x}-2)}\text{dx}=-\int\frac{1}{\text{x}-1}\text{dx}+2\int\frac{1}{\text{x}-2}\text{dx}$
$=-\log|\text{x}-1|+2\log|\text{x}-2|+\text{c}$
$=\log|(\text{x}-2)^2|-\log|\text{x}-1|+\text{c}=\log\Bigg|\frac{(\text{x}-2)^2}{\text{x}-1}\Bigg|+\text{c}$

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