I wander if my thinking is right when solving this equation. I put details in the post. Don't know how to write an equation in the initial message. So when you click on the message you'll see details.

Question

iouri.gordon · Answer

Find $$F \prime \left( X \right)$$ if $$F \left( X \right)=\int\limits_{x}^{x ^{2}}\tan u du$$.
So what I though is by FTC 1 $$F \left( X \right)=F \left( X ^{2} \right)-F \left( X \right)$$, now taking derivative of both sides gives:
$$F \prime \left( X \right) = 2XF \prime \left( X ^{2} \right)-F \prime \left( X \right)$$, which in turn could be rewritten as:
$$2x \frac{ d }{ dx }\int\limits_{0}^{x ^{2}}\tan u du - \frac{ d }{ dx }\int\limits_{0}^{x}\tan udu$$
which in turn  equals:
$$2x \tan x ^{2}-\tan x$$.

Am I doing the right thing?

anonymous · Answer

I dont think my friend ?_? you could take another function G(X) instead taking the same F(X) ...  but why not integrating tan(X) like sin/cos => -sin/-cos => -(sin/-cos) => f'/f => -ln|-cos|

anonymous · Answer

$$\int\limits_{x}^{x^2}\tan(u)du \rightarrow \ln \left( \cos(x) \right) - \ln(\cos(x ^{2}))$$