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Rules of Integration

The area under a curve The Riemann integral Antidifferentiation Rules for integration Integrating $f\left(ax+b\right)$ Particular integrals

Integration by substitution Integration by parts

Extension Problems

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: (do not require Integration by Substitution or Integration by Parts) Don’t forget $+c$

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

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

$\displaystyle \int (\sin^6 x + \cos^6 x + 3\sin^2 x\cos^2 x)\,dx$

Level 2: (may require Integration by Substitution but not Integration by Parts) Don’t forget $+c$

$\displaystyle \int (x+1-e^{-x})\,e^{x e^x}\,dx$