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

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

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

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

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

Solution steps

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

Apply u-substitution

= 2 \cdot \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

= 2 \cdot \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{6} ln ∣ u ∣

Substitute back

u = 3 x^{2} + 5

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

Simplify

2 \cdot \frac{1}{6} ln 3 x^{2} + 5 : \frac{1}{3} ln 3 x^{2} + 5

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

Add a constant to the solution

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

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