What is the integral of x/(sqrt(2x^2+2)) ?

The integral of x/(sqrt(2x^2+2)) is 1/2 sqrt(2x^2+2)+C

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integral of x/(sqrt(2x^2+2))

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

\frac{1}{2} 2 x^{2} + 2 + C

Solution steps

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

Apply u-substitution

= \int \frac{1}{2 2 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{1}{2 u + 2} d u

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

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

Substitute back

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

Simplify

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

= \frac{1}{2} 2 x^{2} + 2

Add a constant to the solution

= \frac{1}{2} 2 x^{2} + 2 + C

What is the integral of x/(sqrt(2x^2+2)) ?
The integral of x/(sqrt(2x^2+2)) is 1/2 sqrt(2x^2+2)+C