Remember that integration is the opposite of differentiation. It's also known as the "antiderivative"

If f(x) = x^3 + 4x, the derivative is 3x^2 + 4

When integrating you will get x^3 + 4x + C

This is how it's done

Take 1 over (exponent +1) times the variable raised to the (n+1) power.  Do this for each term

Thus  we get  3(1/3)x^(2+1) + 4/1(x)^(0 + 1)

= x^3 + 4x

We add a C due to a constant of differentiation