Hi I'm just learning integration, so I'm sorry if this is really basic.
I know this is true: $\displaystyle \int sin^2(x)d x = \frac{1}{2} (x - sin(x)cos(x)) + C$
Because you can use Power reducing substitution http://www.youtube.com/watch?v=VRNuPqA_Vo8
But my question is why can't you use trig substitution like below?
$$\int sin^2x$$ $$= \int \frac{tan^2x}{sec^2x} $$ $$Let\;a = tan x$$ $$As sin^2(x)+cos^2(x)=1...$$ $$= \int \frac{a^2}{1+a^2}$$ $$= 1-tan^-1(a)$$ $$= 1-x$$
Please let me know where I'm assuming something incorrect.