Piecewise representation of $\int x^a \,dx$

Question

I was reading about antiderivative of $\frac{1}{x}$ and they represented it as a piecewise function to include the negatives.

Like this: $$ F(x) = \begin{cases} \ln (-x) + C_1, & x<0 \\ \ln (x) + C_2, &x>0 \end{cases} $$

My question is about the antiderivative of $x^a$ and if you could represent it by including the peices from the antiderivative of $1/x$ along with the integration rule. Is this a correct way of represeting the $ \int x^a \,dx$ ?

$$ \int x^a \,dx = \begin{cases} \frac{x^{a+1}}{a+1} + C_1, & a \neq -1 \\ \ln (-x) + C_2, & a = -1 \wedge (x<0) \\ \ln (x) + C_3, & a = -1 \wedge (x>0) \end{cases} $$

Answer 1

Yes this is correct, I only have two remarks:

• You should define $C_j$. (In your case probably real.)
• You can simply write for $a=1$, $\log\lvert x\rvert+C$, where $C$ might take different values for positive or negative $x$.