What is the integral of (3x)/(x^2+9) ?

The integral of (3x)/(x^2+9) is 3/2 ln|x^2+9|+C

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integral of (3x)/(x^2+9)

\int \frac{3 x}{x ^{2} + 9} d x

\frac{3}{2} ln x^{2} + 9 + C

Solution steps

\int \frac{3 x}{x ^{2} + 9} d x

Take the constant out:

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

= 3 \cdot \int \frac{x}{x ^{2} + 9} d x

Apply u-substitution

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

= 3 \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

= 3 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = x^{2} + 9

= 3 \cdot \frac{1}{2} ln x^{2} + 9

Simplify

3 \cdot \frac{1}{2} ln x^{2} + 9 : \frac{3}{2} ln x^{2} + 9

= \frac{3}{2} ln x^{2} + 9

Add a constant to the solution

= \frac{3}{2} ln x^{2} + 9 + C

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What is the integral of (3x)/(x^2+9) ?
The integral of (3x)/(x^2+9) is 3/2 ln|x^2+9|+C