Find an integral of function

Question

Find an integral of the function:

f(x)= (ln abs(x+7))

abs=absolute value

Answer 1

f(x)=ln|x+7|

Consider what this looks like graphically:

There is a vertical asymptote at x=-7 since ln(0) is undefined.

lim _{x→ -7}+ ln|x+7| = lim _{x→ -7}- ln|x+7|=-∞ and

since ln|±1|=0, (-6,0) and (-8,0) and lastly

lim _{x→ -∞} ln|x+7| = lim _{x→ +∞} ln|x+7|=∞

Now let u=x+7 so that du=dx

f(u)=ln|u|

∫ln(u)du=uln(u)+C=|x+7|ln|x+7|+C