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is differentiable and 
for all 
. Show that for some 
, 
.
such that 
Edit: Proposed by Marian Vasile (congrats!).
where 
where 
where 
![$$
\int_{a^2}^{b^2} \frac{1}{2\sqrt{u}} \, du = \left[ \sqrt{u} \right]_{a^2}^{b^2} = b - a < \frac{1}{a} \quad \text{(since } 0 < a < b \text{)}.
$$](http://latex.artofproblemsolving.com/b/c/2/bc26a37cf63651d0e44daa936743d151cf3a9a3f.png)
is not necessarily true, take 
and 
.
![\[ F(x) = \frac{\cos(x^2)}{2x}. \]](http://latex.artofproblemsolving.com/6/6/9/6697fee3a9fcf0cc1729c29149bf172d3f6fba72.png)
Then, by the product and chain rules,![\[ F'(x) = -\sin(x^2) - \frac{\cos(x^2)}{2x^2}, \]](http://latex.artofproblemsolving.com/9/3/a/93a911bae173c8f8b6d7f7794b496cde9d644660.png)
so that![\[ \sin(x^2) = -F'(x) - \frac{\cos(x^2)}{2x^2}. \]](http://latex.artofproblemsolving.com/d/7/a/d7ac31603c3b785c586adea5930044c2a3c5917d.png)
to 
and applying the FTC gives![\[
\int_a^b \sin(x^2)\,dx
= \left( \frac{\cos(a^2)}{2a} - \frac{\cos(b^2)}{2b} \right)
- \int_a^b \frac{\cos(x^2)}{2x^2}\,dx.
\]](http://latex.artofproblemsolving.com/b/2/8/b281b99866f8bb01264a5fa74b360d0f64536602.png)
and the triangle inequality, we bound![\[
\left| \frac{\cos(a^2)}{2a} - \frac{\cos(b^2)}{2b} \right|
\le \frac{1}{2a} + \frac{1}{2b}
\]](http://latex.artofproblemsolving.com/b/f/c/bfc2953404ef26510c97f6b1081883602734b325.png)
and![\[
\left| \int_a^b \frac{\cos(x^2)}{2x^2}\,dx \right|
\le \int_a^b \frac{1}{2x^2}\,dx
= \frac{1}{2a} - \frac{1}{2b}.
\]](http://latex.artofproblemsolving.com/4/7/b/47ba9906f68337812a9c5483fdce5d39e93fb535.png)
![\[
\left| \int_a^b \sin(x^2)\,dx \right|
\le \frac{1}{2a} + \frac{1}{2b} + \frac{1}{2a} - \frac{1}{2b}
= \frac{1}{a},
\]](http://latex.artofproblemsolving.com/9/6/3/963f18bd32f664664acb7491eaf1f2351dad489a.png)
fails since even a single boundary term, such as 
, can achieve in the worst-case scenario.
2. 
3. 
Bracket 2
2. 
3. 
Bracket 3![$$\int_0^{1}\frac{dx}{1-\sqrt[3]{1-\sqrt{x}}}$$](http://latex.artofproblemsolving.com/d/5/e/d5e8db00985944b2eca341e37a94e34661cb5c86.png)
2. 
3. 
Bracket 4
2. 
3. 
Bracket 5
2. 
3. 
Bracket 6
2. 
3. 
Bracket 7
2. 
3. 
Bracket 8
2. 
3. 
![$$\int_0^{2\pi}\sqrt[4]{\cos^4(x)-\cos(2x)}dx$$](http://latex.artofproblemsolving.com/7/e/e/7eebb4a488fec702e0130a253ef87f0f6dfedb2d.png)
2. ![$$\int_0^1 \frac{x^5}{\sqrt[3]{1-x^3}}dx$$](http://latex.artofproblemsolving.com/6/0/5/605bb9d81d0ccff54a276c9f7bf1c1726c136398.png)
3. 
Bracket 2
2. 
3. 
Bracket 3
2. ![$$\int_0^1 \frac{dx}{(\sqrt{x}+\sqrt[3]{x})^2}$$](http://latex.artofproblemsolving.com/6/9/8/69869c0ed754c0dfacf412ac6411742ffa164a6d.png)
3. 
Bracket 4
2. 
3. 
2. 
3. 
Bracket 2
2. 
3. 
2. 
3. 
2. 
3. 
4. 
5. 
,
and 
.
