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Integrate. Don’t forget $+c$.

$\displaystyle \int{\sin^2{x} dx}=$

$\displaystyle \int \ln x\ dx=$

$\displaystyle \int x^2e^x\ dx=$

$\displaystyle \int{\frac{1}{x\left(\ln x\right)^2}dx}=$

$\displaystyle \int{\frac{\sin{x}}{\cos^3{x}}dx}=$

$\displaystyle \int{\tan x \;dx}=$

$\displaystyle \int{\frac{4 \ln x}{x(1+\left[\ln x\right]^2)}}dx$

$\displaystyle \int{\sin 2x \cos 3x \;dx}$

$\displaystyle \int{x \arctan x \;dx}$

$\displaystyle \int{\frac{dx}{1+e^{-x}}}$

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

Find $\displaystyle \int{x^3\sqrt{4-x^2}\;dx}$
*use the substitution $u=4-x^2$

Find $\displaystyle \int{x^{-\frac{3}{2}}\sqrt{1-x}}\;dx$
*use the substitution $x=\cos^2{\theta}$

The following integrals come from the M.I.T. Integration Bee, an annual competition hosted by the Massachusetts Institute of Technology

You can view some matches here.

Level 1(before Integration by Substitution and Integration by Parts)

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

$\displaystyle \int \frac{\sin^3 x + \sin^2 x - 2\sin x - 2}{\sin^2 x + 2\sin x + 1}\, dx$