Evaluate the following : e^ 4 x -e^ 5 x x^5 d x - Vidyadip

Question

Evaluate the following : $\int \frac{e^{4 \log x}-e^{5 \log x}}{x^5} \cdot d x$

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Answer

$\int \frac{e^{4 \log x}-e^{5 \log x}}{x^5} \cdot d x$
$
\begin{aligned}
& =\int \frac{e^{\log x^4}-e^{\log x^5}}{x^5} \cdot d x, \quad \because a^{\log _a f(x)}=f(x) \\
& =\int\left(\frac{x^4-x^5}{x^5}\right) \cdot d x \\
& =\int\left(\frac{1}{x}-1\right) \cdot d x \\
& =\log (x)-x+c
\end{aligned}
$

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