I am stuck on these three problems because I didn't understand how the teacher was teaching it. The test is in two days.
1.∫(x^3-2x+2)dx
2.∫(sqrt(x) - 1/x^2)dx
3.∫(2x^3+1)^2 x^2 dx
Thanks for the help.
1. $\displaystyle \int{x^3-2x+3}dx=\frac{x^4}{4}-x^2+2x+C$...you do this by basic integration technique...
2.$\displaystyle \int{\sqrt{x}-\frac{1}{x^2}dx}=\int{x^{\frac{1}{2}}-x^{-2}dx}=\frac{2x^{\frac{3}{2}}}{3}+\frac{1}{x}+C$
3.$\displaystyle \int{(2x^3+1)^2\cdot{x^2}}dx=\frac{1}{6}\int{6x^2\ cdot(2x^3+1)^2dx}=\frac{(2x^3+1)^3}{18}+C$
the second one I just converted the variables to a more hospitable form and applied general integration technique..
the second one I just made it so it was an example of $\displaystyle \int{f(g(x))\cdot{g'(x)}}dx$ after doing this I could integrate $\displaystyle f(g(x))$ as though it was just $\displaystyle f(x)$...so by making the derivative of the inside on the outside I could integrate $\displaystyle (2x^3+1)^2$ as though it was just $\displaystyle x^2$ and get $\displaystyle x^3$ and then you substitute back in the $\displaystyle g(x)$
http://www.mathhelpforum.com/math-he...-tutorial.html first of all that will teach you how to use LaTeX and be better understood...now on to the integrals $\displaystyle \int{(2x^3+1)^2\cdot{x}dx}$...for this one honestly I would jsut distribute and integrate...or use the method I will show you on the others
2)$\displaystyle \int{x\sqrt{2x^3+1}dx}$..for this you must make a u-substitution..so you say that $\displaystyle u=\sqrt{2x^3+1}$...then say that $\displaystyle \sqrt[{3}]{\frac{u^2-1}{2}}=x$..and finally then you have that $\displaystyle dx=\frac{\sqrt[{3}]{4}u}{2(u^2-1)^{\frac{2}{3}}}$..and now you can see that this has no means that you could use to find the integral...there are ways but trust me for your level you must have copied it down wrong
3)Ok for this one we have $\displaystyle \int{(x+1)\cdot[x^2+2x-2}]^{\frac{-1}{3}}dx}$which I wont do for you but I will give you a hint $\displaystyle \frac{1}{2}\int{(2x+2)\cdot[x^2+2x-2]^{\frac{-1}{3}}dx}$