What is the integral of x/(sqrt(2x^2+3)) ?

The integral of x/(sqrt(2x^2+3)) is 1/2 sqrt(2x^2+3)+C

integral of x/(sqrt(2x^2+3))

\int \frac{x}{2 x ^{2} + 3} d x

\frac{1}{2} 2 x^{2} + 3 + C

Solution steps

\int \frac{x}{2 x ^{2} + 3} d x

Apply u-substitution

= \int \frac{1}{4 u} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{1}{u} d u

Apply the Power Rule

= \frac{1}{4} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 2 x^{2} + 3

= \frac{1}{4} \cdot 2 (2 x^{2} + 3)^{\frac{1}{2}}

Simplify

\frac{1}{4} \cdot 2 (2 x^{2} + 3)^{\frac{1}{2}} : \frac{1}{2} 2 x^{2} + 3

= \frac{1}{2} 2 x^{2} + 3

Add a constant to the solution

= \frac{1}{2} 2 x^{2} + 3 + C

\frac{d}{d x} (e^{x} - 2 e^{- 2 x})

\int \frac{x ^{2} + 3 x + 7}{x} d x

d er i v a t i v e y = 5 x^{2} e^{2 x}

\frac{d}{d x} (\frac{x ( x ^{2} - 1 )}{x ^{2} + 1})

x \to 0 lim (x^{2} (sin (\frac{1}{x})))

What is the integral of x/(sqrt(2x^2+3)) ?
The integral of x/(sqrt(2x^2+3)) is 1/2 sqrt(2x^2+3)+C