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

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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

= \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

: \int \frac{1}{v} d v

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

Apply the Power Rule

: 2 v^{\frac{1}{2}}

= \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

n \to \infty + lim (1^{n})

x \to - 2 lim (\frac{x ^{3} - 1}{x - 1})

y^{^{''}} + 8 y + 16 y = 0

n = 1 \sum \infty n e^{9 n}

\int (x + 1)^{- 1} d x

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