Tried my best with Latex, took forever!

1)$\displaystyle \int12x(6x^2+1)^3dx$

Let u= $\displaystyle 6x^2+1$

$\displaystyle \frac {du}{dx}=12x$

$\displaystyle du=12xdx$

$\displaystyle \int(u)^312xdx= \int(u)^3du= \frac {u^4}{4} + C$

=$\displaystyle \frac {1}{4}(6x^2+1)^4 + C$

2)$\displaystyle \int \frac {3x^2dx}{\sqrt[3]{x^3+1}}$

Let u= $\displaystyle (x^3+1)$

$\displaystyle du= 3x^2dx$

$\displaystyle \int \frac {du}{\sqrt[3]{u}}$ = $\displaystyle u^{-1/3}du$

=$\displaystyle \frac {u^{2/3}}{2/3} + C$

=$\displaystyle \frac {3}{2}(x^3+1)^{2/3} + C$$\displaystyle \int 2x\cos{(x^2)}dx$

3)

Let u= $\displaystyle (x^2)$ du=$\displaystyle 2xdx$

$\displaystyle

\int \cos{x^2}dx2x= \cos{(u)}du

=\sin{x^2} + C$

**Edit** One more

4)$\displaystyle \int \sin{x}^5 \cos{x}dx$

Answer is (1/6) (sinx) ^6