I have an assignment, which is to show the proofs for

$\displaystyle f(x)=x^{n}$

$\displaystyle F(x)=\frac{x^{n+1}}{n+1}+k$

And also proofs for how you integrate logarithm functions:

$\displaystyle log(x), ln(x)$

And trigonometric functions:

$\displaystyle sin(x), cos(x), tan(x)$

I looked in my books, but they don't have any proofs for why you get what you get when integrating them.