How do you use the second fundamental theorem of Calculus to find the derivative of given #int dx/lnx# from #[x^2, x^3]#?

Answer 1

#= (x ( x - 1 ) )/( ln x)#

you have #d/dx int_(x^2)^(x^3) dx/lnx# which is pretty ugly as the variable x is overloaded, so we can first re-write as
#d/dx int_(x^2)^(x^3) 1/lnt dt qquad square#

The Second FTC asserts, in the most basic terms, that

#d/dx int_a^x f(t) dt = f(x)#

However, since the interval is a function of x in this instance, we must apply the Chain Rule, so:

#color(blue)( d/dx int_a^(u(x)) f(t) dt = f(u(x)) u'(x) )qquad triangle#
Finally, to match #square# to #triangle# we do the following
#int_(x^2)^(x^3) 1/lnt dt #
#= int_(x^2)^(a) 1/lnt dt + int_(a)^(x^3) 1/lnt dt #
#=- int_(a)^(x^2) 1/lnt dt + int_(a)^(x^3) 1/lnt dt #
#= int_(a)^(x^3) 1/lnt dt - int_(a)^(x^2) 1/lnt dt #
and now applying #triangle#!!
#= 1/ln(x^3) d/dx(x^3) - 1/ln(x^2) d/dx(x^2) #
#= (3x^2)/ln(x^3) - (2x)/ln(x^2) #
#= (x ( x - 1 ) )/( ln x)#
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Answer 2

To use the second fundamental theorem of calculus to find the derivative of the integral ( \int_{x^2}^{x^3} \frac{dx}{\ln x} ), first, let ( F(x) ) be any antiderivative of ( \frac{1}{\ln x} ). Then, according to the second fundamental theorem of calculus, the derivative of ( \int_{x^2}^{x^3} \frac{dx}{\ln x} ) with respect to ( x ) is ( F(x^3) \cdot (x^3)' - F(x^2) \cdot (x^2)' ). This simplifies to ( \frac{1}{\ln(x^3)} \cdot 3x^2 - \frac{1}{\ln(x^2)} \cdot 2x ). Finally, using logarithmic properties, this can be rewritten as ( \frac{3x^2}{\ln(x)} - \frac{2x}{\ln(x)} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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