What is the integral of 3/(1-2x) ?

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

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integral of 3/(1-2x)

\int \frac{3}{1 - 2 x} d x

- \frac{3}{2} ln ∣ 1 - 2 x ∣ + C

Solution steps

\int \frac{3}{1 - 2 x} 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}{1 - 2 x} 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 (- \frac{1}{2} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 (- \frac{1}{2} ln ∣ u ∣)

Substitute back

u = 1 - 2 x

= 3 (- \frac{1}{2} ln ∣ 1 - 2 x ∣)

Simplify

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

= - \frac{3}{2} ln ∣ 1 - 2 x ∣

Add a constant to the solution

= - \frac{3}{2} ln ∣ 1 - 2 x ∣ + C

\int cos (2 π x) d x

\frac{\partial}{\partial z} (x y sin (z))

\frac{d y}{d x} = (\frac{x}{2000}) - (\frac{y}{5000}), y (0) = 0

x \to 5 lim (- (\frac{x + 1}{x - 5}))

\int \frac{x}{5} d x