Let F(x) be an antiderivative of (ln x)³/x. If F(1)=0, the F(9)= ?

2 Answers

• ∫(lnx)³/x dx = ∫(lnx)³(1/x) dx

  u = lnx

  du = 1/x dx

  = ∫u³ du = (u^4)/4 + C = ¼[(lnx)^4] + C

  F(1) = 0

  ¼[(ln(1))^4] + C = 0

  ¼(0)^4 + C = 0

  C = 0

  F(x) = ¼[(lnx)^4]

  F(9) = ¼[(ln(9))^4] = 5.827 ; therefore the answer is (c)

  Source(s): Integral Calculator: http://integrals.wolfram.com/index.jsp
  Step-by-Step Derivative Calculator: http://calc101.com/webMathematica/derivatives.jsp
  Another Integral and Derivative Solver: http://www.mathanswerkey.com/solution.php?math=cal

• ∫(lnx)^3 / x dx

  = ∫(lnx)^3 d(lnx)

  = (1/4) (lnx)^4 + c

  F(1) = 0

  => c = 0

  => F(x) = (1/4)(lnx)^4

  => F(9) = (1/4)(ln9)^4

  = 5.827.