Try to solve integral $\int \frac{3-7z}{21z^2-7}dz$

Question

$$\int \frac{3-7z}{21z^2-7}dz$$ I would like to get some advice how to solve this integral

Answer 1

You can use partial fraction decomposition to break the integral down:

$$\int \frac{3-7z}{21z^2-7}\,dz = \int \frac{3- 7z}{7(3z^2 - 1)}\,dz = \int \frac 17\left(\frac{A}{\sqrt{3}z+1} + \frac{B}{\sqrt{3}z-1}\right)\,dz$$

Now just solve for the constants $A, B$: $$A(\sqrt 3 z - 1) + B(\sqrt 3 z + 1) = 3 - 7z$$

Answer 2

You may find the method of Partial Fractions useful to work with the integrand.