Let F(x)= \int\limits^{x^(3)} ___0 ln(t)dt. What is F'(x)?

Question

Let F(x)=
`\int\limits^{x^(3)} ___0` ln(t)dt. What is F'(x)?

Answer 1

9x²ln(x)

Explanation:

Integral of ln(t) with respect to t is:

t×ln(t) - t

Upper limit- lower limit:

x³×ln(x³) - x³ - 0

Note: ln(x³) = 3ln(x)

F(x) = x³ × 3ln(x) - x³

F(x) = 3x³ln(x) - x³ = x³(3ln(x) - 1)

F'(x) = 3x²(3ln(x) - 1) + x³(3/x)

= 3x²(3ln(x) - 1) + 3x²

= 3x²(3ln(x) - 1 + 1)

F'(x) = 9x²ln(x)