What is the integral of x/(5-3x^2) ?

The integral of x/(5-3x^2) is -1/6 ln|5-3x^2|+C

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integral of x/(5-3x^2)

\int \frac{x}{5 - 3 x ^{2}} d x

- \frac{1}{6} ln 5 - 3 x^{2} + C

Solution steps

\int \frac{x}{5 - 3 x ^{2}} d x

Apply u-substitution

: \int - \frac{1}{6 u} d u

= \int - \frac{1}{6 u} d u

Take the constant out:

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

= - \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{6} ln ∣ u ∣

Substitute back

u = 5 - 3 x^{2}

= - \frac{1}{6} ln 5 - 3 x^{2}

Add a constant to the solution

= - \frac{1}{6} ln 5 - 3 x^{2} + C

x d x + \frac{1}{y} d y = 0

\int (e^{- x} - e^{5}) (e^{x} + e^{- 2}) d x

\int x 4^{3 x^{2}} d x

(x y + 2 x + y + 2) d x + (x^{2} + 2 x) d y = 0

\int \frac{x + 1}{x ^{2} - 2 x + 1} d x

What is the integral of x/(5-3x^2) ?
The integral of x/(5-3x^2) is -1/6 ln|5-3x^2|+C