## iouri.gordon 3 years ago 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.

1. iouri.gordon

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?

2. anonymous

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|

3. anonymous

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