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iouri.gordon
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.
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?
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|
\[\int\limits_{x}^{x^2}\tan(u)du \rightarrow \ln \left( \cos(x) \right) - \ln(\cos(x ^{2}))\]