Not all indefinite integrals follow one simple rule. Some are slightly more complicated, but they can be made easier by remembering the derivatives they came from. These complicated indefinite integrals include the integral of a constant (the constant times x), the integral of e^x (e^x) and the integral of x^-1 (ln[x]).
I want to talk a little bit about integral formulas the definite integral formula, the integral of f of x dx equals capital F of x plus c. What it literally means is the antiderivatives of little f of x are capital F of x plus c, so every function of this form would be an antiderivative of little f that's what that formula means. So it's very important to know that every derivative formula has a corresponding integral formula or an equivalent one. So for example this derivative formula derivative of capital F of x equals little f of x corresponds to the integral of little f of x equals capital F of x plus c. Very important that you know that you can go back and forth between these. We've been doing this already we've been using differentiation, to check our integration but you want to be able to come up with integral formulas by starting with a derivative formula and that's what I want to do right now just really quickly.
Since we know that the derivative with respect to x of k times x is k I can turn this around into an integral formula. The integral of kdx=kx plus constant, same thing here since the derivative of e to the x equals e to the x the integral of e to the x, dx equals e to the x plus c and finally this is an important formula the integral of x and negative 1dx this function is ln of the absolute value of x plus c. That's very important to remember when you, for almost every other power you'd use the power rule for antiderivatives, but if that power is negative 1 you've got to use this rule. The integral of x to the negative 1 is natural log, sorry it's a natural log of absolute value of x+c.