int frac{x^{e-1}+e^{x-1}}{x^{e}+e^{x}} d x is | vedclass

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

Question

(C) Let $I = \int \frac{x^{e-1}+e^{x-1}}{x^{e}+e^{x}} d x$. Substitute $t = x^{e}+e^{x}$. Differentiating both sides with respect to $x$,we get $dt =

Vedclass · Accepted Answer

$\frac{1}{e} \log \left(x^{e}+e^{x}\right)+c$

Vedclass · Answer

(C) Let $I = \int \frac{x^{e-1}+e^{x-1}}{x^{e}+e^{x}} d x$. Substitute $t = x^{e}+e^{x}$. Differentiating both sides with respect to $x$,we get $dt = (e x^{e-1} + e^{x-1} \cdot \ln(e)) dx$. Since $\ln(e) = 1$,this simplifies to $dt = e(x^{e-1} + e^{x-1}) dx$. Therefore,$(x^{e-1} + e^{x-1}) dx = \frac{1}{e} dt$. Substituting these into the integral,we get $I = \int \frac{1}{t} \cdot \frac{1}{e} dt = \frac{1}{e} \int \frac{1}{t} dt$. Integrating,we get $I = \frac{1}{e} \log|t| + c$. Substituting back $t = x^{e}+e^{x}$,we get $I = \frac{1}{e} \log(x^{e}+e^{x}) + c$.

$\int \frac{x^{e-1}+e^{x-1}}{x^{e}+e^{x}} d x$ is equal to

Similar Questions

Integrate the function $\frac{1}{x+x \log x}$.

Evaluate the integral: $\int \sec^2 \theta (\sec \theta + \tan \theta)^2 d\theta$

$\int \sin \sqrt{x} \,d x=\ldots+C$ (where $C$ is a constant of integration.)

If $\int \frac{\sin 2x \, dx}{\sin^4 x + \cos^4 x} = \tan^{-1}(f(x)) + c$,then $f\left(\frac{\pi}{3}\right) = $

Integrate the function $\cot x \log \sin x$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module