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dalvyx [7]
3 years ago
15

Finding an Indefinite Integral dx/x(lnx^2)^3

Mathematics
1 answer:
Y_Kistochka [10]3 years ago
3 0

Step-by-step explanation:

Firstly, we'll try to simplify the integrand. By hint 1, we see that:

\ln(x^2) = 2\ln(x)

Simplifying the integrand gives us:

\frac{1}{8}\left(\frac{1}{x(\ln(x))^3}\right)

Next, by hint 2, we observe that:

\frac{d}{dx}\left(\ln(x)\right) = \frac{1}{x}

So this tells us to make the substitution: u = \ln(x)

Doing so gives us:

\int \frac{dx}{x(ln(x^2))^3} = \int \frac{du}{8u^3}, which should be trivial.

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Answer:

Average rate of change = 3

Step-by-step explanation:

  • We know that the average rate of change is basically a measure of the slope of the  secant line  in the closed interval [a, b].

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⇒ Average rate of change = f(b) - f(a)  / b - a

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Therefore, Average rate of change = 3

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