Integration

1) What is ∫x³ dx ?

On Rules of Integration there is a "Power Rule" that says:

∫xⁿ dx = xⁿ⁺¹/(n+1) + C

We can use that rule with n=3:

∫x³ dx = x⁴ /4 + C

2) What is an integral of 2x?

We know that the derivative of x² is 2x ...

... so an integral of 2x is x²

Why must we write +C? 

The derivative of x²+4 is 2x, and the derivative of x²+99 is also 2x, and so on! Because the derivative of a constant is zero.

So when we reverse the operation (to find the integral) we only know 2x, but there could have been a constant of any value.

So we wrap up the idea by just writing + C at the end.

what is ∫cos(x) dx ?

From the Rules of Derivatives table we see the derivative of sin(x) is cos(x) so:

∫cos(x) dx = sin(x) + C