Evaluate the following intregals: 1 x[6( )^2+7 +2] dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{1}{\text{x}[6(\log\text{x})^2+7\log\text{x}+2]}\ \text{dx}$

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Answer

Let $\text{I}=\int\frac{\text{dx}}{\text{x}[6(\log\text{x})^2+7\log\text{x}+2]}$ $=\int\frac{1}{\text{x}(2\log\text{x}+1)(3\log\text{x}+2)}\text{ dx}$ Now,Let $\frac{1}{\text{x}(2\log\text{x}+1)(3\log\text{x}+2)}=\frac{\text{A}}{\text{x}(2\log\text{x}+1)}+\frac{\text{B}}{\text{x}(3\log\text{x}+2)}$
$\Rightarrow1=\text{A}(3\log\text{x}+2)+\text{B}(2\log\text{x}+1)$ Put $\text{x}=10^{-\frac12{}{}}$ $\Rightarrow1=\frac{1}{2}\text{A}\Rightarrow\text{A}=2$ $-\frac{2}{3}$ Put $\text{x}=10)^{-\frac23}$ $\Rightarrow1=-\frac{1}{3}\text{B}\Rightarrow\text{B}=-3$ $\therefore\text{I}=\int\frac{2\text{dx}}{\text{x}(2\log\text{x}+1)}-\int\frac{3\text{dx}}{\text{x}(3\log\text{x}+2)}$ $=\log|2\log\text{x}+1|-\log|3\log\text{x}+2|\text{C}$ $\therefore\text{I}=\log\Big|\frac{2\log\text{x}+1}{3\log\text{x}+2}\Big|+\text{C}$

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