How Do You Solve the Inverse Tangential Integral with Scalar Constants?

[DreamWeaver]

For the scalar constants \(\displaystyle \{a, \, b, \, c,\, d, \, z\} \in \mathbb{R}\), and \(\displaystyle 0<z<1\), find the most general solutions of the parametric integral \(\displaystyle \int_0^z\frac{\tan^{-1}(ax+b)}{(cx+d)}\, dx\)and the restrictions on \(\displaystyle \{a, \, b, \, c, \, d\}\) that satisfy such general solutions.Go on... You know you want to. (Heidy)(Heidy)(Heidy)

 

[Aaron Bex]

The most general solution to the parametric integral is:

\int_0^z\frac{\tan^{-1}(ax+b)}{(cx+d)}\, dx = \frac{1}{c}\ln\left|cx+d\right|\tan^{-1}(ax+b) - \frac{1}{c}\ln\left|c\cdot 0 + d\right|\tan^{-1}(a\cdot 0 + b) + \frac{b}{c}\ln\left|\frac{cz+d}{d}\right|.

The restrictions on \{a, \, b, \, c, \, d\} that satisfy this general solution are:

c ≠ 0 and d ≠ 0.