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Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following intregals:
$\int\frac{\text{ax}^2+\text{bx}+\text{c}}{(\text{x}-\text{a})(\text{x}-\text{b})(\text{x}-\text{c})}\ \text{dx},$ where a, b, c are distinct
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Answer

We have
$\text{I}=\int\frac{\text{ax}^2+\text{bx}+\text{c}}{(\text{x}-\text{a})(\text{x}-\text{b})(\text{x}-\text{c})}\ \text{dx}$
Let $\int\frac{\text{ax}^2+\text{bx}+\text{c}}{(\text{x}-\text{a})(\text{x}-\text{b})(\text{x}-\text{c})}=\frac{\text{A}}{\text{x}-\text{a}}+\frac{\text{B}}{(\text{x}-\text{b}) }+\frac{ \text{C}}{(\text{x}-\text{c})}$
$\Rightarrow\text{ax}^2+\text{bx}+\text{c}\\=\text{A}(\text{x}-\text{b})(\text{x}-\text{c})+\text{B}(\text{x}-\text{c})(\text{x}-\text{a})+\text{C}(\text{x}-\text{a})(\text{x}-\text{b}$
$\Rightarrow\text{ax}^2+\text{bx}+\text{c}\\=\text{A}[\text{x}^2-(\text{b}+\text{c})\text{x}+\text{bc}]+\text{B}[\text{x}^2-(\text{c}+\text{a})\text{x}+\text{ca}]\\+\text{C}[\text{x}^2-(\text{a}-\text{b})\text{x}+\text{ab}]$
$\Rightarrow\text{ax}^2+]\text{bx}+\text{c}=(\text{A}+\text{B}+\text{C})\text{x}^2-[\text{A}(\text{b}+\text{c})+\text{B}(\text{c}+\text{a}) \\+\text{C}(\text{a}+\text{b})]\text{x}+\text{Abc}+\text{Bca}+\text{Cab}$
Equation the coefficient on both sides, we get
$\text{a}=\text{A}+\text{B}+\text{C}\ ...(1)$
$\text{b}=-[\text{A}(\text{b}+\text{c})+\text{B}(\text{c}+\text{a})+\text{C}(\text{a}+\text{b})]\ ...(2)$
$\text{c}=\text{Abc}+\text{Bca}+\text{Cab}\ ...(3)$
Solving (1), (2), (3) we get
$\text{A}=\frac{\text{a}^2+\text{ab}+\text{c}}{(\text{a}-\text{b})(\text{a}-\text{c})}$
$\text{B}=\frac{\text{ab}^2+\text{b}^2+\text{c}^2}{(\text{b}-\text{a})(\text{b}-\text{c})}$
$\text{C}=\frac{\text{ac}^2+\text{bc}+\text{c}}{(\text{c}-\text{a})(\text{c}-\text{b})}$
$\therefore\text{I}=\int\Big[\frac{\text{a}^2+\text{ab}+\text{c}}{(\text{a}-\text{b})(\text{a}-\text{c})}\times\frac{1}{\text{x}-\text{a}}+\frac{\text{ab}^2+\text{b}^2+\text{c}}{(\text{b}-\text{a})(\text{b}-\text{c})}\times\frac{1}{\text{x}-\text{b}}\\+\frac{\text{ac}^2+\text{bc}^2+\text{c}}{(\text{c}-\text{a})(\text{c}-\text{b})}\times\frac{1}{\text{x}-\text{c}}\Big]\ \text{dx}$
$=\frac{\text{a}^2+\text{ab}+\text{c}}{(\text{a}-\text{b})(\text{a}-\text{c})}\log|\text{x}-\text{a}|+\frac{\text{a}^2+\text{ab}+\text{c}}{(\text{a}-\text{b})(\text{a}-\text{c})}\log|\text{x}-\text{b}|\\+\frac{\text{ac}^2+\text{bc}^2+\text{c}}{(\text{c}-\text{a})(\text{c}-\text{b})}\log|\text{x}-\text{c}|+\text{K}$

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