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

For $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$,if $y(x) = \int \frac{\operatorname{cosec} x + \sin x}{\operatorname{cosec} x \sec x + \tan x \sin^2 x} \, dx$ and $\lim_{x \rightarrow (\frac{\pi}{2})^-} y(x) = 0$,then $y\left(\frac{\pi}{4}\right)$ is equal to:

The value of $\int \frac{e^x}{e^x + 1} \,dx$ is

$\int \frac{1+x \cos x}{x\left[1-x^2\left(e^{\sin x}\right)^2\right]} d x=$

$\int \frac{\operatorname{cosec} x}{3 \cos x+4 \sin x} d x=$

If $n$ is a natural number, then $\int \frac{\sin^n x}{\cos^{n+2} x} dx =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module