What is the integral of 3/(0.5-5z) ?

The integral of 3/(0.5-5z) is -0.6ln|0.5-5z|+C

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integral of 3/(0.5-5z)

\int \frac{3}{0.5 - 5 z} d z

- 0.6 ln ∣ 0.5 - 5 z ∣ + C

Solution steps

\int \frac{3}{0.5 - 5 z} d z

Take the constant out:

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

= 3 \cdot \int \frac{1}{0.5 - 5 z} d z

Apply u-substitution

: \int - \frac{1}{5 u} d u

= 3 \cdot \int - \frac{1}{5 u} d u

Take the constant out:

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

= 3 (- \frac{1}{5} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 (- \frac{1}{5} ln ∣ u ∣)

Substitute back

u = 0.5 - 5 z

= 3 (- \frac{1}{5} ln ∣ 0.5 - 5 z ∣)

Simplify

3 (- \frac{1}{5} ln ∣ 0.5 - 5 z ∣) : - 0.6 ln ∣ 0.5 - 5 z ∣

= - 0.6 ln ∣ 0.5 - 5 z ∣

Add a constant to the solution

= - 0.6 ln ∣ 0.5 - 5 z ∣ + C

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\frac{\partial}{\partial x} (\frac{1}{2} (x^{2} - y^{2}))

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x \to 0 lim (x^{2} + 2 x)

What is the integral of 3/(0.5-5z) ?
The integral of 3/(0.5-5z) is -0.6ln|0.5-5z|+C