AP Calculus AB: If f(x) = int[1,x^3] [1/(1+ln(t))] dt, x>=1, find f'(2).

[robmma82]

\(\displaystyle \displaystyle \mbox{If }\, f(x)\, =\, \int_1^{x^3}\, \dfrac{1}{1\, +\, \ln(t)}\, dt\, \mbox{ for }\, x\, \geq\, 1,\)

\(\displaystyle \displaystyle \mbox{then what is the value of }\, f'(2)\,\mbox{?}\)

. . .\(\displaystyle \mbox{(A) }\, \dfrac{1}{1\, +\, \ln(2)}\)

. . .\(\displaystyle \mbox{(B) }\, \dfrac{12}{1\, +\, \ln(2)}\)

. . .\(\displaystyle \mbox{(C) }\, \dfrac{1}{1\, +\, \ln(8)}\)

. . .\(\displaystyle \mbox{(D) }\, \dfrac{12}{1\, +\, \ln(8)}\)

Why D is correct?

 

[ksdhart2]

What are your thoughts? What have you tried? For instance, you first calculated the indefinite integral \(\displaystyle \displaystyle \int \: \frac{1}{1+ln(t)}\) and got... what? Then you evaluated this new function at the points t = x^3 and t = 1, and, using the Fundamental Theorem of Calculushttp://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html, found that the definite integral is... what? Finally, you took the derivative of this function with respect to x and evaluated it at the point x = 2, getting a final answer of... what? Does it match the given answer D?

Please re-read the Read Before Postinghttps://www.freemathhelp.com/forum/threads/41538-Read-Before-Posting!! thread that's stickied at the top of every subforum and comply with its directives, particularly the one about showing your work. Please share with us any and all work you've done on this problem, even the parts you know for sure are wrong. Thank you.

 

[Deleted member 4993]

\(\displaystyle \displaystyle \mbox{If }\, f(x)\, =\, \int_1^{x^3}\, \dfrac{1}{1\, +\, \ln(t)}\, dt\, \mbox{ for }\, x\, \geq\, 1,\)

\(\displaystyle \displaystyle \mbox{then what is the value of }\, f'(2)\,\mbox{?}\)

. . .\(\displaystyle \mbox{(A) }\, \dfrac{1}{1\, +\, \ln(2)}\)

. . .\(\displaystyle \mbox{(B) }\, \dfrac{12}{1\, +\, \ln(2)}\)

. . .\(\displaystyle \mbox{(C) }\, \dfrac{1}{1\, +\, \ln(8)}\)

. . .\(\displaystyle \mbox{(D) }\, \dfrac{12}{1\, +\, \ln(8)}\)

Why D is correct?

Use Leibniz's Rule:

for an integral of the form:

. . .\(\displaystyle \displaystyle \int_{a(x)}^{b(x)}\, f(x,\, t)\, dt\)

then for

. . .\(\displaystyle \displaystyle -\infty\, <\, a(x),\, b(x)\, <\, \infty\)

the derivative of this integral is expressible as

. . .\(\displaystyle \displaystyle \dfrac{\mathrm{d}}{\mathrm{d} x}\, \left(\int_{a(x)}^{b(x)}\, f(x,\, t)\, \mathrm{d}t\right)\)

. . . . .\(\displaystyle \displaystyle =\, f\big(x,\, b(x)\big)\, \cdot\, \dfrac{\mathrm{d}}{\mathrm{d}x}\, b(x)\, -\, f\big(x,\, a(x)\big)\, \cdot\, \dfrac{\mathrm{d}}{\mathrm{d}x}\, a(x)\, +\, \int_{a(x)}^{b(x)}\, \dfrac{\partial}{\partial x}\, f(x,\, t)\, dt\)

Copied from

https://en.wikipedia.org/wiki/Leibniz_integral_rule

 

[robmma82]

Thank you Khan and Hart.