is 
Find the sum of the digits of 
. Prove that
and 
for 
Compute ![\[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\]](http://latex.artofproblemsolving.com/1/9/7/197b63d6a2a31e18dae30ef70595b27318667338.png)
in closed form.
,
.
satisfies the quadratic equation 
and deduce that the value of 
.
.
,
is 
.
an even number.
is not the product of consecutive integers for infinitely many naturals 
.
,
.
be the point on 
such that 
.
.
,
are relatively prime positive integers and 
is a prime number, determine the value of 
.
be the set of all 
matrices with at most two entries in each row equal to 
and all other entries equal to 
.
.
denotes the determinant of the matrix 
.
![\begin{align*}
[(a+1)x-(b-1)y][ax+by]\\
\end{align*}](http://latex.artofproblemsolving.com/b/5/2/b52d93a771b06b5b0a0bebc986de07eda44f25f4.png)
is a polynomial function such that 
then
for rational 
.
Y by centslordm, DeedSpeed, clarkculus, mygoodfriendusesaops, Alex-131, aidan0626, geodash2, abeot, esquire, pinkdino8074, fractalworm, MichelleYMa, Hobz
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 
