1. ## Limits involving integrals

how can i solve by FTC?
$\displaystyle \underset{x\to 0}{\mathop{\lim }}\,\frac {1}{x^{2}}\int_{0}^{x^{2}}{\frac {t + \sin ^{2}\sqrt {t}}{t + 1}}dt$

same condition for this one (use FTC)
find F'(x)
$\displaystyle F\left( x \right) = \int_{0}^{x}{\ln \left( (tx)^{x} \right)}dt$

and this one too :roll:
find
$\displaystyle g''\left( x \right) + g\left( x \right)$
where
$\displaystyle g\left( x \right) = \int_{0}^{x}{f(t)\sin (x)dt}$

2. Originally Posted by vivancos
how can i solve by FTC?
$\displaystyle \underset{x\to 0}{\mathop{\lim }}\,\frac {1}{x^{2}}\int_{0}^{x^{2}}{\frac {t + \sin ^{2}\sqrt {t}}{t + 1}}dt$

same condition for this one (use FTC)
find F'(x)
$\displaystyle F\left( x \right) = \int_{0}^{x}{\ln \left( (tx)^{x} \right)}dt$

and this one too :roll:
find
$\displaystyle g''\left( x \right) + g\left( x \right)$
where
$\displaystyle g\left( x \right) = \int_{0}^{x}{f(t)\sin (x)dt}$
Hint: Apply L'Hopital's rule.

3. Originally Posted by chiph588@
Hint: Apply L'Hopital's rule.
4. $\displaystyle \lim_{x\to 0} \, \frac{\int_{0}^{x^{2}}{\frac {t + \sin ^{2}\sqrt {t}}{t + 1}}dt}{x^2}$
note that direct substitution of $\displaystyle x = 0$ leads to the indeterminate form $\displaystyle \frac{0}{0}$.
$\displaystyle \lim_{x\to 0} \frac{\frac{d}{dx}\left[\int_{0}^{x^{2}}{\frac {t + \sin ^{2}\sqrt {t}}{t + 1}}dt\right]}{\frac{d}{dx}(x^2)} = \lim_{x\to 0} \, \frac{\frac{x^2 + \sin^2{|x|}}{x^2 + 1} \cdot 2x}{2x} = \lim_{x\to 0} \, \frac{x^2 + \sin^2{|x|}}{x^2 + 1} = 0$