knockout 1
1
$$\int\sqrt{12-3x^2}\,dx$$
2
$$\int\frac{x}{1-5x^2}\sqrt{\frac{2}{1+5x^2}-1}\,dx$$
3
$$\int\frac{x^{1010}}{1-x^{2022}}dx$$
4
$$\int_{-\infty}^0x^{2022}e^{4x}dx$$
5
$$\int_0^12^{\left\lfloor \log_{2}{x} \right\rfloor}dx$$
knockout 2
1
$$\int\frac{8x^3}{1+x^8}dx$$
2
$$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\frac{dx}{1-\sin{x}}$$
3
$$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\cos{x}(1+\arctan{x})}{2-\cos^2{x}}dx$$
4
$$\int_2^{\log_2{6}}\frac{2^{2x+1}-9\times2^x-6}{4^x-2^x-6}dx$$
5
$$\int_1^{2022}f(x)dx\,\,,\,\small{f(x) -\left\lfloor x \right\rfloor\text{th decimal place of }x}$$
final
1
$$\int_0^{\infty}\frac{dx}{(x+\frac{1}{x})^2}$$
2
$$\int_{\frac{1}{2}}^1\sum_{k=0}^{\infty}(x^k(x+1)\ln{x}+x^{k^2+k})dx$$
3
$$\int_0^{\infty}\frac{dx}{(1+x^{2022})(1+x^2)}$$
4
$$\int_0^{10}\lim_{s\to x^{+}}(\lim_{t\to \infty}t^{-\frac{|\sin{s}|+\sin{s}}{\sin{s}}})dx$$
5
$$\int_0^{\infty}\sum_{k=\left\lfloor x \right\rfloor}^{\infty}\frac{2^{-k}}{k+1}dx$$
6
$$\int\frac{\sin^3{x}}{\sqrt{\cos{x}}}dx$$
7
$$\int_0^1\frac{(\arcsin{x})^2}{\sqrt{1-x^2}}dx$$
8
$$\int\ln{(1+\sqrt{x})}dx$$
final
1
$$\int\frac{x^2+1}{x\sqrt{x^4-3x^2+1}}dx$$
2
$$\int e^{\sin{x}}(\tan{x}\sec{x}+1)dx$$
3
$$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{x\sin{x}\cos{x}}{e^x+1}dx$$
4
$$\int\frac{1-x^2}{x^4+3x^2+1}dx$$
5
$$;\int_0^1\frac{\tan^{-1}{x}}{1+x}dx$$
6
$$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\tan^{-1}{(e^{\tan{x}})}dx$$
7
$$\int\frac{dx}{\prod_{k=1}^{2021}(x+k)}$$
8
$$\int\frac{dx}{\sin^4{x}+\cos^4{x}}$$
9
$$\int_0^1\frac{x^2-1}{x^3+(1-x^2)^{\frac{3}{2}}}dx$$
10
$$\int e^xx^{e^x-1}(x\ln{x}+1)dx$$
team standoff
1
$$\int\frac{x-1}{x+x^2\ln{x}}dx$$
2
$$\int_0^2\sin^2{\Big(\frac{\pi|x-1|}{2}\Big)}dx$$
3
$$\int_0^{\frac{1}{4}}e^{\sqrt{x}}dx$$
4
$$\int\frac{dx}{\sqrt{x-x^2}}$$
5
$$\int_0^{\frac{\pi}{4}}\ln{(1+\tan{x})}dx$$
6
$$\int_0^{\frac{\pi}{2}}\frac{\cos^2{x}}{\sin{x}+\cos{x}}dx$$
7
$$\int\frac{x^2-1}{x^4+1}dx$$
8
$$\int_{-1}^{1}\sum_{k=0}^{9}kx^kdx$$
9
$$\int_{-1}^{1}\arctan{(2^x)}dx$$
10
$$\int\sqrt{\frac{x}{1-x^3}}dx$$
11
$$\int\frac{dx}{x^2(x^4+1)^{\frac{3}{4}}}$$
12
$$\int_1^{e^2}\frac{\ln{(1+\ln{x})}}{x}dx$$
semifinal
1
$$\int\frac{x^2}{(x\sin{x}+\cos{x})^2}dx$$
2
$$\int_0^1x^m(\ln{x})^ndx\,\,,\,m, n\in \mathbb{Z^+}$$
3
$$\int_{403}^{405}\frac{\sqrt{\ln{(2020-x)}}}{\sqrt{\ln{(2020-x)}}+\sqrt{\ln{(1212+x)}}}dx$$
4
$$\int_0^{\infty}x^{2n}e^{-x^2}dx\,\,,\,n\in\mathbb{Z^+}$$
5
$$\int_{\frac{1}{\pi}}^{\frac{1}{e}}\ln{\left\lfloor \frac{1}{x} \right\rfloor}dx$$
6
$$\int_0^1\sin{(x)}\sinh{(x-1)}dx$$
runner up
1
$$\int_0^1\frac{\sqrt{1-x^2}+\arcsin{\sqrt{\frac{1+x}{2}}}}{\Big|\sin{(\arctan{\frac{\sqrt{1-x^2}}{x}})}\Big|}dx$$
2
$$\int_1^33^{\sqrt{4x-3}}dx$$
final
1
$$\int\frac{x-1}{x+x^2\ln{x}}dx$$
2
$$\int_0^2\sin^2{\Big(\frac{\pi|x-1|}{2}\Big)}dx$$
3
$$\int_0^{\frac{1}{4}}e^{\sqrt{x}}dx$$
4
$$\int\frac{dx}{\sqrt{x-x^2}}$$
5
$$\int_0^{\frac{\pi}{4}}\ln{(1+\tan{x})}dx$$
6
$$\int_0^{\frac{\pi}{2}}\frac{\cos^2{x}}{\sin{x}+\cos{x}}dx$$
7
$$\int\frac{x^2-1}{x^4+1}dx$$
8
$$\int_{-1}^{1}\sum_{k=0}^{9}kx^kdx$$
9
$$\int_{-1}^{1}\arctan{(2^x)}dx$$
10
$$\int\sqrt{\frac{x}{1-x^3}}dx$$
11
$$\int\frac{dx}{x^2(x^4+1)^{\frac{3}{4}}}$$
12
$$\int_1^{e^2}\frac{\ln{(1+\ln{x})}}{x}dx$$
13
$$\int\frac{x}{\sqrt{x}}\frac{\sqrt[3]{x}}{\sqrt[4]{x}}\frac{\sqrt[5]{x}}{\sqrt[6]{x}}\dots dx$$
14
$$\int e^{x^x}\ln{(e^{x^{2x}}x^{x^{2x}})}dx$$
15
$$\int\frac{1}{x}\prod_{k=1}^{\infty}\biggl(1-\tan^2{\Big(\frac{x}{2^k}\Big)}\biggl)dx$$
16
$$\int_0^{\infty}\frac{x}{\sqrt{e^x-1}}dx$$
17
$$\int_0^1x^m(1-x)^ndx$$
$$\int_0^1x^m(1-x)^ndx=\biggl[\frac{x^m(1-x)^n}{m+1}\biggl]_0^1+\frac{n}{m+1}\int_0^1x^{m+1}(1-x)^{n-1}dx$$$$I(m, n)=\frac{n}{m+1}I(m+1, n-1)\text{ or the same thing as }\frac{1}{m+n}I(m+n, 0)=I(m+n-1, 1)$$$$I(m+n, 0)=\frac{1}{m+n+1}\Rightarrow I(m+n-1, 1)=\frac{1}{(m+n)(m+n+1)}\Rightarrow I(m+n-2, 2)=\frac{1\times 2\times 3}{(m+n+1)(m+n)(m+n-1)}$$$$\text{Finally }\,\,\,\,I(m ,n)=\frac{n!}{(m+n+1)(m+n)(m+n-1)\dots(m+1)}=\frac{m!\,n!}{(m+n+1)!}$$
ro16
1
$$\int_{-2}^1\sqrt{e^x}dx$$
2
$$\int_0^1(1+x^2)(1-x^2+x^4-x^6+\dots-x^{4038})dx$$
3
$$\int_{13}^{27}x^2dx$$
4
$$\int\frac{x}{\sqrt{x^2+2x+2}}dx$$
5
$$\int_{-1}^1(x^2+2x^2+3x^3+4x^4+5x^5+6x^6)dx$$
6
$$\int\frac{dx}{\sqrt{3x(4-3x)}}$$
7
$$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{\tan{x}+\text{cot }x}$$
8
$$\int_{-5}^6|x|^3dx$$
quarterfinal
1
$$\int_0^1x^2\sqrt{4-x^2}dx$$
2
$$\int_0^{\pi}e^x\cos^2{(x)}dx$$
3
$$\int_0^1(\text{arccos }x)^2dx$$
4
$$\int_0^{\frac{3\pi}{2}}\text{arccos}(\cos{x})dx$$
semifinal
1
$$\int\frac{x^2}{(x\sin{x}+\cos{x})^2}dx$$
2
$$\int_0^1x^m(\ln{x})^ndx\,\,,\,m, n\in \mathbb{Z^+}$$
3
$$\int_{403}^{405}\frac{\sqrt{\ln{(2020-x)}}}{\sqrt{\ln{(2020-x)}}+\sqrt{\ln{(1212+x)}}}dx$$
4
$$\int_0^{\infty}x^{2n}e^{-x^2}dx\,\,,\,n\in\mathbb{Z^+}$$
5
$$\int_{\frac{1}{\pi}}^{\frac{1}{e}}\ln{\left\lfloor \frac{1}{x} \right\rfloor}dx$$
6
$$\int_0^1\sin{(x)}\sinh{(x-1)}dx$$
third place
1
$$\int_{-1}^{1}\sin{(\pi|x|)}\arcsin{\big(\sqrt{x}\big)}dx$$
2
$$\int\frac{e^{2x}+2e^x+1}{e^{2x}-2e^{x}+1}dx$$
final
1
$$\int_0^{2019\pi}\sum_{k=0}^5\sin^{-1}{(\sin{kx})}dx$$
2
$$\int\frac{70\sin{x}+23\cos{x}}{5\sin{x}+8\cos{x}}dx$$
3
$$\int_{-\frac{\pi}{4040}}^{\frac{\pi}{4040}}\frac{\cos^{2020}{2020x}}{(2020^{2020x}+1)(\sin^{2020}{2020x}+\cos^{2020}{2020x})}dx$$
4
$$\int\sqrt{x}e^{\sqrt{x}}dx$$
5
$$\int e^{2019x+e^{2019x}}dx$$
6
$$\int_{20}^{89}3x^2dx$$