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  4. If ∫ (e5 log e x-e4 log e x/e3 log e x-e2 log e x) d x=(x3/λ)+c, then find λ

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Q. If $\int \frac{e^{5 \log _{e} x}-e^{4 \log _{e} x}}{e^{3 \log _{e} x}-e^{2 \log _{e} x}} d x=\frac{x^{3}}{\lambda}+c$, then find $\lambda$

Integrals

Solution:

$\int \frac{e^{5 \log _{e} x}-e^{4 \log _{e} x}}{e^{3 \log _{e} x}-e^{2 \log _{e} x}} d x$
$= \int \frac{x^{5}-x^{4}}{x^{3}-x^{2}} d x$
$=\int \frac{x^{2}\left(x^{3}-x^{2}\right)}{x^{3}-x^{2}} d x$
$=\int x^{2} d x$
$=\frac{x^{3}}{3}+C$