I want to find
$$\int1+\sin^2x+\sin^4x+\sin^6x +...\text{dx}$$
My method was to interpret the integrand as a geometric series with first term $1$ and common ratio $\sin^2x$. Assuming $\sin x\ne1$, I reasoned the integrand should converge.

So the integrand should be equal to
$$\frac{1}{1-\sin^2x}=\frac{1}{\cos^2x}=\sec^2{x}$$
$$\implies \int1+\sin^2x+\sin^4x+\sin^6x +...\text{dx}=\int\sec^2{x}~\text{dx}=\tan x$$
I would just like to know, is this right?
If it is, it seems to me a rather beautiful result.