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

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

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integral of x/(x^2+5)

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

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

Solution steps

\int \frac{x}{x ^{2} + 5} 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 = x^{2} + 5

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

Add a constant to the solution

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

\frac{d y}{d x} = e^{4 x} + 7 y

\int \frac{7 x ^{3} + 3 x}{x - 1} d x

\int 3 e^{- 2 x} d x

\frac{2 x ^{4}}{cos ( x ) e ^{x}}

x \to 3 + lim (\frac{∣ x - 3 ∣}{x - 3})

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