The correct option is A a/b [ x log ( b+ce^x ) ]. nbsp;I=∫ nbsp; a / b+ce^ x dx Substitute t= e ^ x ⇒ dt= e ^ x dx nbsp;I=∫ a / t( b+ct ...

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Functions

∫a/b+cexdx.

Question

$\int \frac{a}{b + c e^{x}} d x .$

\frac{a}{b} [x - log (b + c e^{x})] .

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\frac{a}{b} [x + log (b + c e^{x})] .

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\frac{a}{b} [x - log (c e^{x})] .

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\frac{a}{c} [x + log (b + c e^{x})] .

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Solution

The correct option is A $\frac{a}{b} [x - log (b + c e^{x})] .$
$I = \int \frac{a}{b + c e^{x}} d x$
Substitute $t = e^{x} \Rightarrow d t = e^{x} d x$
$I = \int \frac{a}{t (b + c t)} d t = a \int (\frac{1}{b t} - \frac{c}{b^{2} + b c t}) d t$
$= - \frac{a}{b} log (b^{2} + b c t) + \frac{a}{b} t$
$= a \frac{x - log (b + c e^{x})}{b}$

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