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

The integral of x/(2x^2+3) is 1/4 ln|2x^2+3|+C

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

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

\frac{1}{4} ln 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

= \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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{4} ln ∣ u ∣

Substitute back

u = 2 x^{2} + 3

= \frac{1}{4} ln 2 x^{2} + 3

Add a constant to the solution

= \frac{1}{4} ln 2 x^{2} + 3 + C

7 + e^{- 5 x} y^{^{'}} = 0

\int \frac{2 x - 1}{( x - 1 ) ( x - 2 )} d x

sin (w)

y^{^{''}} + 4 y^{^{'}} + 5 y = 15 x + e^{(- x)}

y^{^{'}} = 2 x e^{(2 y)}

What is the integral of x/(2x^2+3) ?
The integral of x/(2x^2+3) is 1/4 ln|2x^2+3|+C