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

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

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integral of x/(5-x^2)

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{1}{2 u} d u

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{2} ln ∣ u ∣

Substitute back

u = 5 - x^{2}

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

Add a constant to the solution

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

\int arcsin (u) d u

\int_{0}^{x} 1 d x

\frac{d}{d x} ((5 x + 1)^{2})

f (θ) = 3 sec (θ)

\int \frac{( x ^{2} + 3 x - 1 )}{x} d x

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