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

\int_{3}^{- 1} x^{2} d x

x \to - 3 lim (\frac{\frac{3}{x} + 1}{x + 3})

\frac{d y}{d x} = 2 y + x^{2} + 5

y^{^{'}} + \frac{3}{2} y = 1 - \frac{1}{3} t

\frac{d}{d x} ((3 x^{2} - 9)^{- 13})