Derivatives and Integration of absolute value functions

The proofs below will use the absolute value functions form as

$$|f(x)|=\sqrt{f(x)^2}$$

Claim 1:  $${d\over dx}|x|={x\over |x|}$$
Proof:

Claim 2:  $${d\over dx}|f(x)|={f(x)\over |f(x)|}f'(x)$$

Proof:

Claim 3: $$\int |x| dx={x|x|\over 2}+c$$

Proof:

Claim 4: $$\int x|x| dx={x^2|x|\over 3}+c$$

Proof:

Yes it does generalise,

Claim 5: $$\int x^n|x| dx={x^{n+1}|x|\over n+2}+c$$
Proof:

Claim 6: $$\int |f(x)| dx=??$$

Proof: