int frac{a , dx}{b + c e^x} = | vedclass

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Question

(A) Let $I = \int \frac{a \, dx}{b + c e^x}$.Multiply numerator and denominator by $e^{-x}$:$I = \int \frac{a e^{-x} \, dx}{b e^{-x} + c}$.Let $u = b

Vedclass · Accepted Answer

$\frac{a}{b} \log \left( \frac{e^x}{b + c e^x} \right) + C$

Vedclass · Answer

(A) Let $I = \int \frac{a \, dx}{b + c e^x}$.Multiply numerator and denominator by $e^{-x}$:$I = \int \frac{a e^{-x} \, dx}{b e^{-x} + c}$.Let $u = b e^{-x} + c$,then $du = -b e^{-x} \, dx$,which implies $e^{-x} \, dx = -\frac{du}{b}$.Substituting these into the integral:$I = a \int \frac{-du/b}{u} = -\frac{a}{b} \int \frac{du}{u} = -\frac{a}{b} \log |u| + C$.Substituting back $u = b e^{-x} + c = \frac{b + c e^x}{e^x}$:$I = -\frac{a}{b} \log \left| \frac{b + c e^x}{e^x} \right| + C = \frac{a}{b} \log \left| \frac{e^x}{b + c e^x} \right| + C$.

$\int \frac{a \, dx}{b + c e^x} = $

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