Consider the following problem:

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}tanxdx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{sinx}{cosx}dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}sinx*secxdx$

Let $u=secx$ and $dv=sinxdx$

$du=secxtanxdx$ and $v=-cosx$

Using the integration $by$ parts formula: $uv-{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$vdu $we$ get:

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}tanxdx=secx*(-cosx)-{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(-cosx)(secxtanx)dx$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}tanxdx=-1+{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}tanxdx$

$0=-1$

There's a flaw $in$ reasoning here. Explain.