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

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

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integral of (-3)/(3x+2)

\int \frac{- 3}{3 x + 2} d x

- ln ∣ 3 x + 2 ∣ + C

Solution steps

\int \frac{- 3}{3 x + 2} d x

Take the constant out:

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

= - 3 \cdot \int \frac{1}{3 x + 2} d x

Apply u-substitution

: \int \frac{1}{3 u} d u

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 3 \cdot \frac{1}{3} ln ∣ u ∣

Substitute back

u = 3 x + 2

= - 3 \cdot \frac{1}{3} ln ∣ 3 x + 2 ∣

Simplify

- 3 \cdot \frac{1}{3} ln ∣ 3 x + 2 ∣ : - ln ∣ 3 x + 2 ∣

= - ln ∣ 3 x + 2 ∣

Add a constant to the solution

= - ln ∣ 3 x + 2 ∣ + C

\int - \frac{1}{t ^{3}} d t

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

\int (\frac{1}{x ^{2} + 5 x + 3}) d x

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

\int \frac{2 x}{3 x - 2} d x

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