Find f if f′(x)=2/x,x<0 and f(−1)=3.
Find f when f"(x) = x^-2, x > 0, f(1) = 0, f(2) = 0
Integrate twice. Do not forget the constants of integration.
f'(x) = -1/x + C
f(x) = - ln x + Cx + D
To find C and D, apply f(1) and f(2)
f(1) = - ln (1) + C + D = 0
C + D = 0 ........... equation 1
f(2) = - ln 2 + 2C + D = 0
2C + D = ln 2 .............. equation 2
Solve for C and D from equations 1 and 2
C = ln 2
D = - ln2
f(x) = - ln x + (ln 2) x - ln 2