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

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

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integral of (2x)/(x^2+5)

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

ln x^{2} + 5 + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 2 \cdot \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = x^{2} + 5

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

Simplify

2 \cdot \frac{1}{2} ln x^{2} + 5 : ln x^{2} + 5

= ln x^{2} + 5

Add a constant to the solution

= ln x^{2} + 5 + C

\int_{0}^{2} \frac{2 x}{( x ^{2} + 5 ) ^{2}} d x

\frac{d}{d x} ((ln (x))^{\frac{1}{2}})

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

\int e^{- t} \cdot (- t^{2} + t - 1) d t

\int_{0}^{4} (2 + x x) d x

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