Solve ∫ wrt z/sqrt{a-bz}

$$\int \frac { d z } { \sqrt { a - b } z }$$

Evaluate

$$\frac{\ln(|z|)}{\sqrt{a-b}}+С$$

Solution Steps

Factor out the constant using \int af\left(z\right)\mathrm{d}z=a\int f\left(z\right)\mathrm{d}z.

$$\frac{\int \frac{1}{z}\mathrm{d}z}{\sqrt{a-b}}$$

Use \int \frac{1}{z}\mathrm{d}z=\ln(|z|) from the table of common integrals to obtain the result.

$$\frac{\ln(|z|)}{\sqrt{a-b}}$$

If F\left(z\right) is an antiderivative of f\left(z\right), then the set of all antiderivatives of f\left(z\right) is given by F\left(z\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{\ln(|z|)}{\sqrt{a-b}}+С$$

Differentiate w.r.t. a

$$-\frac{\ln(|z|)}{2\left(a-b\right)^{\frac{3}{2}}}$$