What is the integral of (xe^x)/(sqrt(e^x-1)) ?

The integral of (xe^x)/(sqrt(e^x-1)) is 2xsqrt(e^x-1)-4(-arctan(sqrt(e^x-1))+sqrt(e^x-1))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (xe^x)/(sqrt(e^x-1))

\int \frac{x e ^{x}}{e ^{x} - 1} d x

2 x e^{x} - 1 - 4 (- arctan (e^{x} - 1) + e^{x} - 1) + C

Solution steps

\int \frac{x e ^{x}}{e ^{x} - 1} d x

Apply u-substitution

: \int \frac{ln ( u )}{u - 1} d u

= \int \frac{ln ( u )}{u - 1} d u

Apply Integration By Parts

: 2 ln (u) u - 1 - \int \frac{2 u - 1}{u} d u

= 2 ln (u) u - 1 - \int \frac{2 u - 1}{u} d u

\int \frac{2 u - 1}{u} d u = 4 (- arctan (u - 1) + u - 1)

= 2 ln (u) u - 1 - 4 (- arctan (u - 1) + u - 1)

Substitute back

u = e^{x}

= 2 ln (e^{x}) e^{x} - 1 - 4 (- arctan (e^{x} - 1) + e^{x} - 1)

Simplify

ln (e^{x}) : x

= 2 x e^{x} - 1 - 4 (- arctan (e^{x} - 1) + e^{x} - 1)

Add a constant to the solution

= 2 x e^{x} - 1 - 4 (- arctan (e^{x} - 1) + e^{x} - 1) + C

\int \frac{ln ( tan ( x ) )}{cos ( x ) sin ( x )} d x

\int x^{2} cos^{2} (\frac{x}{2}) d x

What is the integral of (xe^x)/(sqrt(e^x-1)) ?
The integral of (xe^x)/(sqrt(e^x-1)) is 2xsqrt(e^x-1)-4(-arctan(sqrt(e^x-1))+sqrt(e^x-1))+C