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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS3 Marks
Question
Evaluate the following:
$\int\frac{\text{e}^{6\log\text{x}}-\text{e}^{5\log\text{x}}}{\text{e}^{4\log\text{x}}-\text{e}^{3\log\text{x}}}\text{dx}$
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Answer

Let $\text{I}=\int\frac{\text{e}^{6\log\text{x}}-\text{e}^{5\log\text{x}}}{\text{e}^{4\log\text{x}}-\text{e}^{3\log\text{x}}}\text{dx}$
$=\int\frac{\text{e}^{\log\text{x}^6}-\text{e}^{\log\text{x}^5}}{\text{e}^{\log\text{x}^4}-\text{e}^{3\log\text{x}^3}}\text{dx}$ $\big[\because\ \text{a}\log\text{b}-\text{b}-\log\text{b}^{\text{a}}\big]$
$=\int\frac{\text{x}^6-\text{x}^5}{\text{x}^4-\text{x}^3}\text{dx}$ $\big[\because\ \text{e}^{\log\text{x}}=\text{x}\big]$
$=\int\frac{\text{x}^3-\text{x}^2}{\text{x}-1}\text{dx}=\int\frac{\text{x}^2(\text{x}-1)}{\text{x}-1}\text{dx}$ $=\int\text{x}^2\text{dx}=\frac{\text{x}^3}{3}+\text{C}$

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