$\displaystyle \int\frac{xsinx}{1+(cosx)^{2}}dx$

$\displaystyle \int_{-\infty }^{\infty }e^{-x^{2}}dx$

$\displaystyle \int\frac{\sqrt{\sqrt{x^4 + 1}-x^{2}}}{x^4 + 1}dx$

$\displaystyle \int\frac{\sqrt{x}ln(x)e^{x}x^{3}}{\sqrt{lnx}}dx$

I have been working on these integral for 4 weeks... haven't been able to get anything yet. I am preparing for a competition and came across these bad boys.