Olympiad problems

2021 CMIMC Integration Bee, 11

Tags: calculus , integration , CMIMC , integration bee

$$\int_0^\frac{\pi}{2}\frac{1}{4-3\cos^2(x)}\,dx$$ [i]Proposed by Connor Gordon[/i]

Tags: CMIMC , integration bee

$$\int\frac{\ln 2}{1+2^{-x}}\,dx$$ [i]Proposed by Connor Gordon[/i]

Tags: calculus , integration , integration bee

\[\int_{\pi/4}^{\pi/2} \tan^{-1}\left(\tan^2(x)\right)\sin(2x)\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

Tags: calculus , integration , CMIMC , integration bee

$$\int_{-\infty}^\infty\frac{x\arctan(x)}{x^4+1}\,dx$$ [i]Proposed by Connor Gordon[/i]

Tags: calculus , integration , integration bee

\[\int_{-1}^1 \sqrt[3]{x}\log(1+e^x)\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Tags: calculus , integration , integration bee

\[\int_0^1 \frac{1-x}{x^{5/2}+x^{3/2}+x^{1/2}}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Tags: CMIMC , integration bee

$$\int_1^2\left(x^5+6x^4+14x^3+16x^2+9x+2\right)dx$$ [i]Proposed by Connor Gordon[/i]

Tags: calculus , integration , CMIMC , integration bee

$$\int_1^\infty \frac{1 + 2x \ln 2}{x\sqrt{x 4^x - 1}}\,dx$$ [i]Proposed by Vlad Oleksenko[/i]

Tags: calculus , integration , integration bee

\[\int_1^e \frac{2x^2+1}{x^3+x\log(x)}\mathrm dx\] [i]Proposed by Connor Gordon[/i]