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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

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} = $

Similar Questions

$\int \sec^p x \tan x \, dx = $

If $f(x) = \int \frac{16x^7 + 5x^{10}}{(x^3 + 2 + 3x^8)^2} dx$ for $x \geq 0$ and $f(0) = 1$, then the value of $f(1)$ is

To evaluate $\int x^3 e^{3x^2 + 5} dx$,the simplest way is to

$\int \frac{d x}{(x+2) \sqrt{x+1}} = $

$\int \sqrt{4 \cos ^2 x - 5 \sin ^2 x} \cos x \, dx =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module