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Home  >>  CBSE XII  >>  Math  >>  Integrals
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Evaluate the following$\large\int\frac{e^{6log x}-e^{5log x}}{e^{4log x}-e^{3log x}}$$dx$

$\begin{array}{1 1} (A)\;\frac{x^2}{2}\\ (B)\;\frac{x^3}{3} \\ (C)\; x \\ (D)\;log x\end{array} $

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1 Answer

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Toolbox:
  • $\large e^{logx}=x$
  • $\int x^n dx=\large\frac{x^{n+1}}{n+1}+c$
Let $I=\Large\int\frac{e^{6log x}-e^{5log x}}{e^{4log x}-e^{3log x}}$$dx$
This can be written as,
$\Large \int\frac{e^{log x^6}-e^{log x^5}}{e^{log x^4}-e^{log x^3}}$$dx$
But $\large e^{logx}=x$
Hence $I=\large\frac{x^6-x^5}{x^4-x^3}$$dx$
Taking $x^5$ as the common factor in the numerator and $x^3$ as the common factor in the denominator, we get
$I=\int\large\frac{x^5(x-1)}{x^3(x-1)}$$dx$
$=\int x^2 dx$
on integrating we get,
$I=\large\frac{x^3}{3}+c$
answered Apr 11, 2013 by meena.p
edited Feb 6, 2014 by rvidyagovindarajan_1
 
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