be a non constant polynomial with real coefficients. Prove that the system of simultaneous equations —
has finitely many solutions 
Write only the answers to the questions given.
to be more exact:
in years 2002-08 time was 90' for part A and 120' for part B
red balls and 
white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...
unit. The area of the hexagon is ...
and 
.
is ...
so that 
and 
is ...
,
,
,
has an average of 
and a median of 
.
are integers greater than 
which are four consecutive quarters of an arithmetic row with 
.
and 
are squares of two consecutive natural numbers, then the smallest value of 
is ...
,
,
and 
.
and 
lies on the line segment 
.
and 
.
is ...
meet 
for each natural number 
.
is ....
provided that the difference of any two numbers at least 
is ...
which satisfies 
are as many as ...
and 
.
.
is a point on the side extension 
so that 
is perpendicular to 
and 
.
is ...
consists of 
with 
positive reals is ....
with 
is ....
square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of 
coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of 
if 
students, 
students play cricket, 
students play hockey and 
students play basketball. 
students play both basketball and hockey, 
students play both cricket and basketball, 
students play both cricket and hockey, and 
students play all three: basketball, hockey, and cricket. Find the number of students who do not play any game.
such that 
divides 
.
of 
such that 
.
,
,
,
has area ![$\tfrac{3}{4}[ABC]$](http://latex.artofproblemsolving.com/9/8/a/98a118cb3595f9ea91f8b38b7293cc211f81295a.png)
.
be a permutation of 
such that there are exactly 
satisfying 
and 
.
Y by centslordm, DeedSpeed, clarkculus, mygoodfriendusesaops, Alex-131, aidan0626, geodash2, abeot, esquire, pinkdino8074, fractalworm
Hello AoPS!
Along with user geodash2, I have organized another high-school integration bee (River Hill High School Spring Integration Bee) and wanted to share the problems!
We had enough folks for two concurrent rooms, hence the two sets. (ARML kids from across the county came.)
Keep in mind that these integrals were written for a high-school contest-math audience. I hope you find them enjoyable and insightful; enjoy!
![\[
\int_{1}^{2} \frac{x^{3}+x^2}{x^5}dx
\]](//latex.artofproblemsolving.com/5/a/b/5ab9b8014b74dbfcf4db15873e0348bbdc0171ef.png)
![\[\int_{2025}^{2025^{2025}}\frac{1}{\ln\left(2025\right)\cdot x}dx\]](//latex.artofproblemsolving.com/9/f/3/9f3fe22d62ad9fc5259d97fcf60eb3c45d240bfa.png)
![\[
\int(\sin^2(x)+\cos^2(x)+\sec^2(x)+\csc^2(x))dx
\]](//latex.artofproblemsolving.com/8/3/d/83da36cded674b594ce7d585b1c1fee7df4956ae.png)
![\[
\int_{-2025.2025}^{2025.2025}\sin^{2025}(2025x)\cos^{2025}(2025x)dx
\]](//latex.artofproblemsolving.com/9/1/0/91045d7c16160403507ee5f66a7cd021dfe0a0c3.png)
![\[
\int_{\frac \pi 6}^{\frac \pi 3} \tan(\theta)^2d\theta
\]](//latex.artofproblemsolving.com/3/4/f/34f4936db8bf49e9d0c77de8539ba6ee144f94d5.png)
![\[
\int \frac{1+\sqrt{t}}{1+t}dt
\]](//latex.artofproblemsolving.com/b/4/b/b4bead66ef90e5677281987993f82b1e4bb864df.png)
![\[\int \frac{x^{2}+2x+1}{x^{3}+3x^{2}+3x+3}dx
\]](//latex.artofproblemsolving.com/9/2/2/922c2e35f18adffdd453ad926d0ae91fe759dd50.png)
![\[\int_{0}^{\frac{3\pi}{2}}\left(\frac{\pi}{2}-x\right)\sin\left(x\right)dx\]](//latex.artofproblemsolving.com/3/6/b/36b02db24ba60cb50e0027a1d0d8a3ce1d07e72f.png)
![\[
\int_{-\pi/2}^{\pi/2}x^3e^{-x^2}\cos(x^2)\sin^2(x)dx
\]](//latex.artofproblemsolving.com/a/c/5/ac50d60b5ff6d26d8ef2079efaaf7e2d293c8798.png)
![\[
\int\frac{1}{\sqrt{12-t^{2}+4t}}dt
\]](//latex.artofproblemsolving.com/9/0/d/90d94cd0cdb89832dbf6b91dfb2af29169005963.png)
![\[
\int \frac{\sqrt{e^{8x}}}{e^{8x}-1}dx
\]](//latex.artofproblemsolving.com/1/c/0/1c091415a17d162a745bd79dfb6c1632297d9ca0.png)
![\[
\int \frac{e^x}{e^{2x}+1} dx
\]](//latex.artofproblemsolving.com/c/9/c/c9c93d4a2fc4e8856e0b6d30cffd5bf2581521bf.png)
![\[
\int_{-5}^5\sqrt{25-u^2}du
\]](//latex.artofproblemsolving.com/7/6/0/760ca61e6e609ada7052eddb31c0b5cc8bdc7d8a.png)
![\[
\int_{-\frac12}^\frac121+x+x^2+x^3\ldots dx
\]](//latex.artofproblemsolving.com/f/7/a/f7a1aa77b71a36c1800ed7bf5e44fa6744ee9a04.png)
![\[\int \cos(\cos(\cos(\ln \theta)))\sin(\cos(\ln \theta))\sin(\ln \theta)\frac{1}{\theta}d\theta\]](//latex.artofproblemsolving.com/e/0/3/e030efdb1f1d73326dd09a60c961ac4b7bf987df.png)
![\[\int_{0}^{\frac{1}{6}}\frac{8^{2x}}{64^{2x}-8^{\left(2x+\frac{1}{3}\right)}+2}dx\]](//latex.artofproblemsolving.com/4/a/2/4a274f4254113198782619e28ac63edcb41664ff.png)
![\[\int_{0}^{\frac{\pi}{2}}\frac{\sin\left(x\right)}{\sin\left(x\right)+\cos\left(x\right)}+\frac{\sin\left(\frac{\pi}{2}-x\right)}{\sin\left(\frac{\pi}{2}-x\right)+\cos\left(\frac{\pi}{2}-x\right)}dx\]](//latex.artofproblemsolving.com/2/0/5/2051d51ec5ec892ce1f571385bfc1f74d05bea9f.png)
![\[
\int_0^{\infty}e^{-x}\Bigl(\cos(20x)+\sin(20x)\Bigr) dx
\]](//latex.artofproblemsolving.com/4/e/4/4e482f5c117a0f3984c822614dd4a4214cc03b86.png)
![\[
\lim_{n\to \infty}\frac{1}{n}\int_{1}^{n}\sin(nt)^2dt
\]](//latex.artofproblemsolving.com/3/b/3/3b3bf68b93dbe49e9979471ba04b82c3a2eb96c4.png)
![\[
\int_{x=0}^{x=1}\left( \int_{y=-x}^{y=x} \frac{y^2}{x^2+y^2}dy\right)dx
\]](//latex.artofproblemsolving.com/c/0/3/c0314f295bc45fa87cfffc93ee348f38b44f51b2.png)
![\[
\int_{0}^{13}\left\lceil\log_{10}\left(2^{\lceil x\rceil }x\right)\right\rceil dx
\]](//latex.artofproblemsolving.com/4/f/0/4f012045fcbc8210ea752729f609149b8a534e4f.png)
![\[
\int \frac{6x^2}{x^6+2x^3+2}dx
\]](//latex.artofproblemsolving.com/0/d/1/0d1131a496881bcf7036de0b2a36191008925840.png)
![\[
\int -\sin(2\theta)\cos(\theta)d\theta
\]](//latex.artofproblemsolving.com/b/0/0/b006ec3908ed9d9c00fe354578d50b5f91d56ddb.png)
![\[
\int_{0}^{5}\sin(\frac{\pi}2 \lfloor{x}\rfloor x) dx
\]](//latex.artofproblemsolving.com/a/2/1/a21381f8fa2432165c02e7cd28dfb00ae3ad5475.png)
![\[
\int_{0}^{1} \frac{\sin^{-1}(\sqrt{x})^2}{\sqrt{x-x^2}}dx
\]](//latex.artofproblemsolving.com/b/c/b/bcbe046b36c5d072bd6e064ee98a94683ccadfcc.png)
![\[
\int\left(\cot(\theta)+\tan(\theta)\right)^2\cot(2\theta)^{100}d\theta
\]](//latex.artofproblemsolving.com/0/1/2/012796576e56002d2983da33436ddd6208ae3507.png)
![\[
\int \ln\left\{\sqrt[7]{x}^\frac1{\ln\left\{\sqrt[5]{x}^\frac1{\ln\left\{\sqrt[3]{x}^\frac1{\ln\left\{\sqrt{x}\right\}}\right\}}\right\}}\right\}dx
\]](//latex.artofproblemsolving.com/d/6/f/d6fa1f1a12b988fef90f2a39f9b77f6e9822f76c.png)
![\[\int_{1}^{{e}^{\pi}} \cos(\ln(\sqrt{u}))du\]](//latex.artofproblemsolving.com/e/1/5/e152dacb1601c772f32ab9f3494d45bed43c635f.png)
![\[
\int_e^{\infty}\frac {1-x\ln{x}}{xe^x}dx
\]](//latex.artofproblemsolving.com/4/e/8/4e8ae310d63fae062122d6ff9ba1570ffaf98a42.png)
![\[\int_{0}^{1}\frac{e^{x}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{1}}}}\times\frac{e^{-\frac{x^{2}}{2}}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{2}}}}\times\frac{e^{\frac{x^{3}}{3}}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{3}}}}\times\frac{e^{-\frac{x^{4}}{4}}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{4}}}} \ldots \,dx\]](//latex.artofproblemsolving.com/4/2/f/42feb97860007e1ef6ce36e03c71743b253f2576.png)
For
on the domain 
it is known that ![\[\displaystyle f(x)=\sin\left(\int_{0}^x \sqrt[3]{\cos\left(\frac{\pi}{2} t\right)^3+26}\ dt\right)\]](//latex.artofproblemsolving.com/d/0/a/d0af1de2e29589d6205f05bad4cf0412742535b8.png)
is invertible. What is 
?
Along with user geodash2, I have organized another high-school integration bee (River Hill High School Spring Integration Bee) and wanted to share the problems!
We had enough folks for two concurrent rooms, hence the two sets. (ARML kids from across the county came.)
Keep in mind that these integrals were written for a high-school contest-math audience. I hope you find them enjoyable and insightful; enjoy!
Warm Up Problems
![\[
\int_{1}^{2} \frac{x^{3}+x^2}{x^5}dx
\]](http://latex.artofproblemsolving.com/5/a/b/5ab9b8014b74dbfcf4db15873e0348bbdc0171ef.png)
![\[\int_{2025}^{2025^{2025}}\frac{1}{\ln\left(2025\right)\cdot x}dx\]](http://latex.artofproblemsolving.com/9/f/3/9f3fe22d62ad9fc5259d97fcf60eb3c45d240bfa.png)
![\[
\int(\sin^2(x)+\cos^2(x)+\sec^2(x)+\csc^2(x))dx
\]](http://latex.artofproblemsolving.com/8/3/d/83da36cded674b594ce7d585b1c1fee7df4956ae.png)
![\[
\int_{-2025.2025}^{2025.2025}\sin^{2025}(2025x)\cos^{2025}(2025x)dx
\]](http://latex.artofproblemsolving.com/9/1/0/91045d7c16160403507ee5f66a7cd021dfe0a0c3.png)
![\[
\int_{\frac \pi 6}^{\frac \pi 3} \tan(\theta)^2d\theta
\]](http://latex.artofproblemsolving.com/3/4/f/34f4936db8bf49e9d0c77de8539ba6ee144f94d5.png)
![\[
\int \frac{1+\sqrt{t}}{1+t}dt
\]](http://latex.artofproblemsolving.com/b/4/b/b4bead66ef90e5677281987993f82b1e4bb864df.png)
Easier Division Set 1
![\[\int \frac{x^{2}+2x+1}{x^{3}+3x^{2}+3x+3}dx
\]](http://latex.artofproblemsolving.com/9/2/2/922c2e35f18adffdd453ad926d0ae91fe759dd50.png)
![\[\int_{0}^{\frac{3\pi}{2}}\left(\frac{\pi}{2}-x\right)\sin\left(x\right)dx\]](http://latex.artofproblemsolving.com/3/6/b/36b02db24ba60cb50e0027a1d0d8a3ce1d07e72f.png)
![\[
\int_{-\pi/2}^{\pi/2}x^3e^{-x^2}\cos(x^2)\sin^2(x)dx
\]](http://latex.artofproblemsolving.com/a/c/5/ac50d60b5ff6d26d8ef2079efaaf7e2d293c8798.png)
![\[
\int\frac{1}{\sqrt{12-t^{2}+4t}}dt
\]](http://latex.artofproblemsolving.com/9/0/d/90d94cd0cdb89832dbf6b91dfb2af29169005963.png)
![\[
\int \frac{\sqrt{e^{8x}}}{e^{8x}-1}dx
\]](http://latex.artofproblemsolving.com/1/c/0/1c091415a17d162a745bd79dfb6c1632297d9ca0.png)
Easier Division Set 2
![\[
\int \frac{e^x}{e^{2x}+1} dx
\]](http://latex.artofproblemsolving.com/c/9/c/c9c93d4a2fc4e8856e0b6d30cffd5bf2581521bf.png)
![\[
\int_{-5}^5\sqrt{25-u^2}du
\]](http://latex.artofproblemsolving.com/7/6/0/760ca61e6e609ada7052eddb31c0b5cc8bdc7d8a.png)
![\[
\int_{-\frac12}^\frac121+x+x^2+x^3\ldots dx
\]](http://latex.artofproblemsolving.com/f/7/a/f7a1aa77b71a36c1800ed7bf5e44fa6744ee9a04.png)
![\[\int \cos(\cos(\cos(\ln \theta)))\sin(\cos(\ln \theta))\sin(\ln \theta)\frac{1}{\theta}d\theta\]](http://latex.artofproblemsolving.com/e/0/3/e030efdb1f1d73326dd09a60c961ac4b7bf987df.png)
![\[\int_{0}^{\frac{1}{6}}\frac{8^{2x}}{64^{2x}-8^{\left(2x+\frac{1}{3}\right)}+2}dx\]](http://latex.artofproblemsolving.com/4/a/2/4a274f4254113198782619e28ac63edcb41664ff.png)
Harder Division Set 1
![\[\int_{0}^{\frac{\pi}{2}}\frac{\sin\left(x\right)}{\sin\left(x\right)+\cos\left(x\right)}+\frac{\sin\left(\frac{\pi}{2}-x\right)}{\sin\left(\frac{\pi}{2}-x\right)+\cos\left(\frac{\pi}{2}-x\right)}dx\]](http://latex.artofproblemsolving.com/2/0/5/2051d51ec5ec892ce1f571385bfc1f74d05bea9f.png)
![\[
\int_0^{\infty}e^{-x}\Bigl(\cos(20x)+\sin(20x)\Bigr) dx
\]](http://latex.artofproblemsolving.com/4/e/4/4e482f5c117a0f3984c822614dd4a4214cc03b86.png)
![\[
\lim_{n\to \infty}\frac{1}{n}\int_{1}^{n}\sin(nt)^2dt
\]](http://latex.artofproblemsolving.com/3/b/3/3b3bf68b93dbe49e9979471ba04b82c3a2eb96c4.png)
![\[
\int_{x=0}^{x=1}\left( \int_{y=-x}^{y=x} \frac{y^2}{x^2+y^2}dy\right)dx
\]](http://latex.artofproblemsolving.com/c/0/3/c0314f295bc45fa87cfffc93ee348f38b44f51b2.png)
![\[
\int_{0}^{13}\left\lceil\log_{10}\left(2^{\lceil x\rceil }x\right)\right\rceil dx
\]](http://latex.artofproblemsolving.com/4/f/0/4f012045fcbc8210ea752729f609149b8a534e4f.png)
Harder Division Set 2
![\[
\int \frac{6x^2}{x^6+2x^3+2}dx
\]](http://latex.artofproblemsolving.com/0/d/1/0d1131a496881bcf7036de0b2a36191008925840.png)
![\[
\int -\sin(2\theta)\cos(\theta)d\theta
\]](http://latex.artofproblemsolving.com/b/0/0/b006ec3908ed9d9c00fe354578d50b5f91d56ddb.png)
![\[
\int_{0}^{5}\sin(\frac{\pi}2 \lfloor{x}\rfloor x) dx
\]](http://latex.artofproblemsolving.com/a/2/1/a21381f8fa2432165c02e7cd28dfb00ae3ad5475.png)
![\[
\int_{0}^{1} \frac{\sin^{-1}(\sqrt{x})^2}{\sqrt{x-x^2}}dx
\]](http://latex.artofproblemsolving.com/b/c/b/bcbe046b36c5d072bd6e064ee98a94683ccadfcc.png)
![\[
\int\left(\cot(\theta)+\tan(\theta)\right)^2\cot(2\theta)^{100}d\theta
\]](http://latex.artofproblemsolving.com/0/1/2/012796576e56002d2983da33436ddd6208ae3507.png)
Bonanza Round (ie Fun/Hard/Weird Problems) (In No Particular Order)
![\[
\int \ln\left\{\sqrt[7]{x}^\frac1{\ln\left\{\sqrt[5]{x}^\frac1{\ln\left\{\sqrt[3]{x}^\frac1{\ln\left\{\sqrt{x}\right\}}\right\}}\right\}}\right\}dx
\]](http://latex.artofproblemsolving.com/d/6/f/d6fa1f1a12b988fef90f2a39f9b77f6e9822f76c.png)
![\[\int_{1}^{{e}^{\pi}} \cos(\ln(\sqrt{u}))du\]](http://latex.artofproblemsolving.com/e/1/5/e152dacb1601c772f32ab9f3494d45bed43c635f.png)
![\[
\int_e^{\infty}\frac {1-x\ln{x}}{xe^x}dx
\]](http://latex.artofproblemsolving.com/4/e/8/4e8ae310d63fae062122d6ff9ba1570ffaf98a42.png)
![\[\int_{0}^{1}\frac{e^{x}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{1}}}}\times\frac{e^{-\frac{x^{2}}{2}}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{2}}}}\times\frac{e^{\frac{x^{3}}{3}}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{3}}}}\times\frac{e^{-\frac{x^{4}}{4}}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{4}}}} \ldots \,dx\]](http://latex.artofproblemsolving.com/4/2/f/42feb97860007e1ef6ce36e03c71743b253f2576.png)
For
on the domain 
it is known that ![\[\displaystyle f(x)=\sin\left(\int_{0}^x \sqrt[3]{\cos\left(\frac{\pi}{2} t\right)^3+26}\ dt\right)\]](http://latex.artofproblemsolving.com/d/0/a/d0af1de2e29589d6205f05bad4cf0412742535b8.png)
is invertible. What is 
?This post has been edited 1 time. Last edited by Riemann123, Apr 11, 2025, 2:14 PM
Reason: Forgot a problem
Reason: Forgot a problem
![$f(x)=\sin\left(\int_{0}^x \sqrt[3]{\cos\left(\frac{\pi}{2} t\right)^3+26}\ dt\right)$](http://latex.artofproblemsolving.com/8/9/e/89eeee4193af4b9c470e81e73b6310bdd9b4f291.png)
,
and ![$f'(x)=\cos\left(\int_{0}^x \sqrt[3]{\cos\left(\frac{\pi}{2} t\right)^3+26}\ dt\right)\sqrt[3]{\cos\left(\frac{\pi}{2} x\right)^3+26}$](http://latex.artofproblemsolving.com/b/c/c/bccb6109fd0a9b70b1150194a36a6787053275c1.png)
.
in that interval, Then by inverse function theorem: ![$(f^{-1})'(x)=\dfrac{1}{f'(f^{-1}(x)}=\dfrac{1}{f'(0)}=\dfrac{1}{\cos 0\times \sqrt[3]{1+26}}=\dfrac{1}{3}$](http://latex.artofproblemsolving.com/e/4/1/e41ecfef71f4d5cd163d3d176011d0b3b7eb5ad4.png)
![\[\int \ln\left\{\sqrt[7]{x}^\frac1{\ln\left\{\sqrt[5]{x}^\frac1{\ln\left\{\sqrt[3]{x}^\frac1{\ln\left\{\sqrt{x}\right\}}\right\}}\right\}}\right\}dx=\int\frac{\frac17\ln x\frac13\ln x}{\frac15\ln x\frac12\ln x}dx=\boxed{\frac{10}{21}x+C}\]](http://latex.artofproblemsolving.com/7/3/7/7371ac7bceb97c2ae349e045671dc4ec5e259aad.png)
,
,
.![\[\int_{1}^{{e}^{\pi}} \cos(\ln(\sqrt{u}))du=2\int_0^{\pi/2}e^{2w}\cos w dw\]](http://latex.artofproblemsolving.com/8/9/e/89ec3ffcf66af3175b6365ffd078e414f3d06ebf.png)
This is a well known cyclic IBP.![\[=\frac25(e^{2w}\sin w+2e^{2w}\cos w)\bigg|_0^{\pi/2}=\boxed{\frac{2e^\pi-4}{5}}\]](http://latex.artofproblemsolving.com/e/8/9/e892cf8727fd14ccc8479a549f1ef451d4338511.png)
![\[
\int_e^{\infty}\frac {1-x\ln{x}}{xe^x}dx=\int_e^{\infty}x^{-1}e^{-x}-\ln{x}e^{-x}dx=e^{-x}\ln{x}\bigg|_{e}^\infty=\boxed{-e^{-e}}
\]](http://latex.artofproblemsolving.com/6/9/0/69002d8d65d26bade02487829e6bf171abcc6299.png)
and the bottom one is 1.![\[\int_{0}^{1}\frac{e^{x}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{1}}}}\times\frac{e^{-\frac{x^{2}}{2}}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{2}}}}\times\frac{e^{\frac{x^{3}}{3}}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{3}}}}\times\frac{e^{-\frac{x^{4}}{4}}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{4}}}} \ldots \,dx=\int_0^1\frac{x+1}{(x+1)(x+2)}dx\]](http://latex.artofproblemsolving.com/5/f/7/5f7070d4a7b4bd94678d86a53b748938f264bd15.png)
![\[=\int_0^1\frac{1}{x+2}=\ln(x+2)|_0^1=\boxed{\ln3-\ln2}\]](http://latex.artofproblemsolving.com/4/9/2/492ed9971b868707a82df3630f9ea92a41d09bf8.png)
![\[\int_{1}^{2} \frac{x^{3}+x^2}{x^5}dx=-x^{-1}-\frac12 x^{-2}\bigg|_1^2=\boxed{\frac78}\]](http://latex.artofproblemsolving.com/3/7/4/374da2f50a7edc9733a8dc858afa4a43ef8301db.png)
![\[\int_{2025}^{2025^{2025}}\frac{1}{\ln\left(2025\right)\cdot x}dx=\frac{\ln(x)}{\ln(2025)}\bigg|_{2025}^{2025^{2025}}=\boxed{2024}\]](http://latex.artofproblemsolving.com/5/8/e/58eccff45bef5597ce2f57e432ef202d7f889872.png)
![\[\int(\sin^2(x)+\cos^2(x)+\sec^2(x)+\csc^2(x))dx=\boxed{x+\tan(x)-\cot(x)+C}\]](http://latex.artofproblemsolving.com/b/2/3/b23a82ca7401e3ad73c1bd5a14986ecdf130ded5.png)
![\[\int_{-2025.2025}^{2025.2025}\sin^{2025}(2025x)\cos^{2025}(2025x)dx=\boxed{0}\]](http://latex.artofproblemsolving.com/6/7/4/6746b3fa76d852472d181844e3d06a770c85f638.png)
![\[\int_{\frac \pi 6}^{\frac \pi 3} \tan(\theta)^2d\theta=\tan\theta-\theta\bigg|_{\pi/6}^{\pi/3}=\boxed{\frac{2\sqrt3}{3}-\frac{\pi}{6}}\]](http://latex.artofproblemsolving.com/7/7/2/77241c849517866a6d56a2e302f14beb8e376c9b.png)
and 
.![\[
\int \frac{1+\sqrt{t}}{1+t}dt=2\int\frac{u+u^2}{1+u^2}du=\ln|1+u^2|+2\int1-\frac{1}{1+u^2}du=\boxed{\ln|1+t|+2\sqrt{t}-2\tan^{-1}(\sqrt{t})+C}\]](http://latex.artofproblemsolving.com/a/8/0/a8099ad1c55d97ab95d19feab44f455f8a4d6a0e.png)
\]
then IBP.![\[\int_{0}^{\frac{3\pi}{2}}\left(\frac{\pi}{2}-x\right)\sin\left(x\right)dx=\int_{-\pi}^{\pi/2}u\cos u du=u\sin u+\cos u\bigg|_{-\pi}^{\pi/2}=\boxed{\frac\pi2+1}\]](http://latex.artofproblemsolving.com/5/d/7/5d7012acc0d0b6e9a022d510398a5e9cb0a2d345.png)
![\[
\int_{-\pi/2}^{\pi/2}x^3e^{-x^2}\cos(x^2)\sin^2(x)dx=\boxed{0}
\]](http://latex.artofproblemsolving.com/c/d/3/cd3c63ba6e376818da56ccc52314bb64065292a6.png)
![\[
\int\frac{1}{\sqrt{12-t^{2}+4t}}dt=\int\frac{1}{\sqrt{4^2-(t-2)^2}}dt=\boxed{\sin_{-1}\left(\frac{t-2}{4}\right)+C}
\]](http://latex.artofproblemsolving.com/2/2/8/228733156f5f1bee23de9518430e8e0b43587a90.png)
,
.![\[\int \frac{\sqrt{e^{8x}}}{e^{8x}-1}dx=\frac14\int\frac{du}{u^2-1}=\boxed{\frac18\ln\left|\frac{e^{4x}-1}{e^{4x}+1}\right|+C}\]](http://latex.artofproblemsolving.com/a/d/1/ad130be7625506a32c110f887b003c167dd97ede.png)
by u-sub.
.![\[\int \frac{e^x}{e^{2x}+1} dx=\int\frac{du}{u^2+1}=\boxed{\tan^{-1}(e^x)+C}\]](http://latex.artofproblemsolving.com/b/d/5/bd5a3190bd5d4ba5b02cc1e1a5ac7dd8dd583e37.png)
![\[
\int_{-5}^5\sqrt{25-u^2}du=\boxed{\frac{25\pi}{2}}
\]](http://latex.artofproblemsolving.com/e/8/7/e871b0488189a82e23647e0df5d458facaae6546.png)
![\[
\int_{-\frac12}^\frac121+x+x^2+x^3\ldots dx=\int_{-\frac12}^\frac12 \frac{dx}{1-x}=-\ln|1-x|\bigg|_{-\frac12}^\frac12=\boxed{\ln3}
\]](http://latex.artofproblemsolving.com/c/e/4/ce4d85eec296daf75c9a4bd2f5386f36caec0a1a.png)
![\[\int \cos(\cos(\cos(\ln \theta)))\sin(\cos(\ln \theta))\sin(\ln \theta)\frac{1}{\theta}d\theta=\boxed{\sin(\cos(\cos(\ln \theta)))+C}\]](http://latex.artofproblemsolving.com/9/e/0/9e0200331d713ca454b72b5eb5ec7684f6386706.png)
![\[\int_{0}^{\frac{1}{6}}\frac{8^{2x}}{64^{2x}-8^{\left(2x+\frac{1}{3}\right)}+2}dx=\int_{0}^{\frac{1}{6}}\frac{64^{x}}{64^{2x}-(2)64^{x}+2}dx\]](http://latex.artofproblemsolving.com/5/3/e/53eb332eff80af1421ddb0c6e9bf295bb66cd7ed.png)
,
![\[=\frac{1}{\ln(64)}\int_1^2\frac{du}{(u-1)^2+1}=\frac{1}{\ln(64)}\tan^{-1}(u-1)\bigg|_1^2=\boxed{\frac{\pi}{24\ln(2)}}\]](http://latex.artofproblemsolving.com/f/8/4/f84b0e67aa21f2fd1a6acd7947ace127741f3b05.png)
This is equal to 
