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

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

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

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

- ln ∣ 2 x - 3 ∣ + C

Solution steps

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

Take the constant out:

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

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

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

Simplify

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

= - ln ∣ 2 x - 3 ∣

Add a constant to the solution

= - ln ∣ 2 x - 3 ∣ + C

n = 1 \sum \infty 5^{n} x^{n}

arctan (4 x)

\frac{d}{d x} (e^{x} + sin (x))

\int \frac{u}{u} d u

\frac{d y}{d x} + 4 y = 9

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