Indefinite integral of a simple function

Question

$\int 2(1 + \tan^2 x)$

My work :

$2(1 + \tan^2(x) = 2 + 2\tan^2x$

$2x + \frac{2}{3}$ $\tan^3(x) \cdot \ln|sec(x)| + C$

The answer says no, after multiple tries :(

Answer 1

$$\int2(1+\tan^2x)dx=2\int\sec^2x\ dx=2\tan x+K$$ as $$\frac{d(\tan x)}{dx}=\sec^2x$$

Answer 2

$$\sec^2 x-1=\tan^2 x$$

$$2\int \sec^2 x-1\,\mathrm{d}x+2\int \mathrm{d}x=2\int \sec^2 x\,\mathrm{d}x=2\tan x+c$$

Answer 3

Your mistake here was in thinking that

$$\int(\tan x)^2 dx = \frac{(\tan x)^3}{3} + C\tag{wrong!}$$

but this is not correct. The rule that

$$\int x^a dx = \frac{x^{a+1}}{a+1} + C$$

only works when $x$ is literally the same variable that appears in $dx$.

This does actually allow you to write things like

$$\int (\tan x)^a d(\tan x) = \frac{(\tan x)^{a+1}}{a+1} + C$$

but you have to make sure you understand what $d(\tan x)$ means, and you have to be careful because this doesn't work if the thing in the differential (here $\tan x$) is not sufficiently "well-behaved" throughout the region of integration.