What is the integral of (xe^{x^2})/(sqrt(e^{x^2)+2)} ?

The integral of (xe^{x^2})/(sqrt(e^{x^2)+2)} is sqrt(e^{x^2)+2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (xe^{x^2})/(sqrt(e^{x^2)+2)}

\int \frac{x e ^{x^{2}}}{e ^{x^{2}} + 2} d x

e^{x^{2}} + 2 + C

Solution steps

\int \frac{x e ^{x^{2}}}{e ^{x^{2}} + 2} d x

Apply u-substitution

= \int \frac{e ^{u}}{2 e ^{u} + 2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{e ^{u}}{e ^{u} + 2} d u

Apply u-substitution

= \frac{1}{2} \cdot \int \frac{1}{v} d v

Apply the Power Rule

= \frac{1}{2} \cdot 2 v^{\frac{1}{2}}

Substitute back

= \frac{1}{2} \cdot 2 (e^{x^{2}} + 2)^{\frac{1}{2}}

Simplify

\frac{1}{2} \cdot 2 (e^{x^{2}} + 2)^{\frac{1}{2}} : e^{x^{2}} + 2

= e^{x^{2}} + 2

Add a constant to the solution

= e^{x^{2}} + 2 + C

What is the integral of (xe^{x^2})/(sqrt(e^{x^2)+2)} ?
The integral of (xe^{x^2})/(sqrt(e^{x^2)+2)} is sqrt(e^{x^2)+2}+C