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

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Question

(A) Let $I = \int \frac{e^x dx}{\sqrt{a + b e^x}}$.Substitute $a + b e^x = t$. Then,differentiating both sides with respect to $x$,we get $b e^x dx =

Vedclass · Accepted Answer

$\frac{2}{b} \sqrt{a + b e^x} + c$

Vedclass · Answer

(A) Let $I = \int \frac{e^x dx}{\sqrt{a + b e^x}}$.Substitute $a + b e^x = t$. Then,differentiating both sides with respect to $x$,we get $b e^x dx = dt$,which implies $e^x dx = \frac{1}{b} dt$.Substituting these into the integral,we get:$I = \int \frac{1}{\sqrt{t}} \cdot \frac{1}{b} dt = \frac{1}{b} \int t^{-1/2} dt$.Using the power rule for integration,$\int t^n dt = \frac{t^{n+1}}{n+1} + c$,we have:$I = \frac{1}{b} \cdot \frac{t^{1/2}}{1/2} + c = \frac{2}{b} \sqrt{t} + c$.Substituting $t = a + b e^x$ back,we get:$I = \frac{2}{b} \sqrt{a + b e^x} + c$.

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

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