Question

Find the derivative with respect to x of the integral from 2 to x^3 of ln(x^2)dx

Answer 1

The derivative follows from the fundamental theorem of calculus:

`\displaystyle(\mathrm d)/(\mathrm dx)\int_c^(g(x))f(t)\,\mathrm dt=f(g(x))(\mathrm dg)/(\mathrm dx)`

where
`c` is any constant in the domain of
`f`.

We have

`g(x)=x^3\implies(\mathrm dg)/(\mathrm dx)=3x^2`

so

`\displaystyle(\mathrm d)/(\mathrm dx)\int_2^(x^3)\ln(t^2)\,\mathrm dt=3x^2\ln((x^3)^2)=18x^2\ln x`

(applying the property
`\ln a^b=b\ln a`)