Illinois Bee Qualification
1
$$\int_{\pi^e}^{e^{\pi}}1^xdx$$
2
$$\text{If }\int_{\varphi e}^{10\pi}\frac{\sin{x}}{x}dx=A, \frac{dA}{dx}=\text{?}$$
3
$$\int\frac{61}{60\sin{x}+11\cos{x}}dx$$
4
$$\int\frac{dx}{x^{1729}+x}$$
5
$$\int\sin{(\sin{(\sin(\dots(x)))})}dx$$
6
$$\int\frac{2000x^{2018}-18}{x^{2019}+x}dx$$
7
$$\int\frac{x^{-\frac{7}{12}}}{\sqrt[3]{x}+\sqrt[4]{x}}dx$$
8
$$\int\frac{\sec{(x-A)}}{\sqrt{1-\sin^2{(x-B)}}}dx$$
9
$$\int\text{csc}^3(x)\sec{(x)}dx$$
10
$$\int\frac{x\ln{x}\text{sech}^2(x)-\tanh{(x)}}{x\ln^2{x}}dx$$
11
$$\text{Prove }\int e^{nix}=\frac{1}{n}[\sin{(nx)}-i\cos{(nx)}]\,,\,\forall n \in \mathbb{N}$$
12
$$g(x)=\sum_{k=1}^n[kx^{k-2}-\frac{h'(x)}{(k+1)x^2}]$$
$$h(x)=x^{k+1}$$
$$1+\int[g(x)dx]=\text{?}$$
13
$$\int_3^7\frac{\ln{(x+2)}}{\ln{(24+10x-x^2)}}dx$$
14
$$\int_0^1\frac{(x-x^2)^4}{1+x^2}dx$$
15
$$\int_{-\infty}^{\infty}e^{b+at-t^2}dt$$
1
$$\int_0^1x(1-x)^{2015}dx$$
2
$$\int\frac{dx}{x^2+4x+5}$$
3
$$\int_0^{\frac{\pi}{2}}\sin^2{(x)}dx$$
4
$$\int\frac{e^{\arctan{(2x)}}}{5+20x^2}dx$$
5
$$\int\frac{3x-2}{x^2+1}dx$$
6
$$\int\cos{(\sqrt{x})}dx$$
7
$$\int \big(\ln{(3x^2+6x+3)}-2\ln{(x+1)}\big)dx$$
8
$$\int_{-1}^{1}\frac{\,\,\,\,\,\,\,\,\,\,x^{2015}}{\sqrt[2015]{1-x}\,\,\,\,+\sqrt[2015]{1+x}}\,\,\,dx$$
9
$$\int\frac{dx}{x^2\sqrt{x^2+25}}$$
10
$$\int\frac{e^{2x}}{1+e^{4x}}dx$$
11
$$\int_0^{\pi}\cos^4{(x)}dx$$
12
$$\int_{-2015}^{2015}\left\lfloor x \right\rfloor dx$$
13
$$\int_0^1\frac{x+1}{\sqrt[3]{x}+1}dx$$
14
$$\int_1^7\sqrt{(7-x)(x-1)}\,dx$$
15
$$\int_0^1\ln{(x^2+1)}dx$$
16
$$\int\biggl(\frac{1}{\ln{x}}-\frac{1}{\ln^2{x}}\biggl)dx$$
17
$$\int_0^{\infty}2^{-\left\lfloor x \right\rfloor}dx$$
18
$$\int(1-4x^2)e^{-2x^2}dx$$
19
$$\int\frac{5\sin{x}}{\cos{x}+2\sin{x}}dx$$
20
$$\int\frac{x+\sin{x}-\cos{x}-1}{x+e^x+\sin{x}}\,dx$$