Assignment 19

course Mth 272

7/28 10:30 am

019.

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Question: `qQuery problem 6.4.16 use table to integrate x^2 ( ln(x^3) )^2

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Your solution: We can first simplify the expression before taking the integral using u and du. We find that u = x^3 and du = 3x^2. We can now rewrite our expression as 1/3 Int(3x^2(ln(x^3))^2, and then as 1/3 Int(du(ln(u))^2. We find that integral 43 from the table of integrals works in this situation. When we apply that formula, we end up with u(lnu)^2 – 2 Int(lnu)du. When we substitute back in for u, we get 1/3(x^3(ln(x^3))^2 – 2(1/(x^3))).

confidence rating: 3

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Given Solution:

`a Let u = x^3 so du = 3x^2 dx. The x^2 dx in the integral is just 1/3 du.

You therefore have the integral of 1/3 ( ln(u) )^2 du.

The table should have something for ( ln(u) ) ^ n.

In any case the integral of ln(u)^2 with respect to u is u ln(u)^2 - 2 u ln(u) + 2 u.

With the substitution u = x^3 you would be integrating (ln u)^2 * du/3, which would give you

u [ 2 - 2 ln u + (ln u)^2 ] / 3, which translates to

x^3 ( 2 - 2 ln(x^3) + (ln(x^3) ) ^ 2 ) / 3.

DER: int( (ln(u)^2) = u•LN(u)^2 - 2•u•LN(u) + 2•u. Then for increasing powers of n int( ln(u)^n) gives us:

u•LN(u)^3 - 3•u•LN(u)^2 + 6•u•LN(u) - 6•u then

u•LN(u)^4 - 4•u•LN(u)^3 + 12•u•LN(u)^2 - 24•u•LN(u) + 24•u and

u•LN(u)^5 - 5•u•LN(u)^4 + 20•u•LN(u)^3 - 60•u•LN(u)^2 + 120•u•LN(u) - 120•

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Self-critique (if necessary):

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Self-critique Rating:

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Question: `qQuery problem 6.4.46 use table to integrate x ^ 4 ln(x) then check by integration by parts

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Your solution: To simplify this expression before we integrate it, we can use integration by parts. We find that u = lnx and du = 1/x. Dv = x^n and v = x^(n+1)/(n+1). So using the rule, our new integral is ln(x)(x^(n+1)/(n+1)) - Int(x^(n+1)/(n+1))/x), which simplifies to x^(n+1)/((n+1)(ln(x)-1(n+1))). Since n=4 in our original expression, we can plug 4 in for n to solve for the final integral. So it becomes x^(4+1)/((4+1)(ln(x)-1(4+1))) = x^5 /(5ln(x) – 5).

confidence rating: 3

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Given Solution:

`a Integration by parts on x^n ln(x) works with the substitution

u = ln(x) and dv = x^n dx, so that

du/dx = 1/x and v = x^(n+1) / n, giving us

du = dx / x and v = x^(n+1) / (n + 1).

Thus our integral is

u v - int( v du) =

ln(x) * x^(n+1) / (n + 1) - int ( x^(n+1)/(n+1) * dx / x) =

ln(x) * x^(n+1)/( n + 1) - int(x^n dx) / (n+1) =

ln(x) * x^(n+1) / (n + 1) - x^(n+1) / (n+1)^2 =

x^(n+1) / (n+1) ( ln(x) - 1(n+1)).

This should be equivalent to the formula given in the text.

For n = 4 we get

x^(4 + 1) / (4 + 1) ( ln(x) - 1 / (4 + 1)) =

x^5 / 5 (ln(x) - 1/5). *&*&

(x^5/25)(4 ln x) + C

Using integration by parts:

(x^5/5) ln x - (x^5/25)

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Self-critique (if necessary):

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Self-critique Rating:

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Question: `qQuery problem 6.4.63 profit function P = `sqrt( 375.6 t^2 - 715.86) on [8,16].

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Your solution: Because we know that the function is made up of u^2 and a^2, we can find u and a and use those quantities to find the average profit. We find that u^2 = 375.6t^2 and that a^2 = 715.86. By taking the square roots of both of those, we find that u = sqrt(375.6)t = 19.38t and a = 26.755. Now we can find the profit from 8 to 16, which we find to be about 1850. Since 16-8 = 8, we can divide 1850 by 8 to get the average profit, which we find to be about 230.

confidence rating:3

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Given Solution:

`a To get the average net profit integrate the profit function over the given interval and divide by the length of the interval.

The integrand is sqrt(375.6 t^2 - 715.86), which is of the form `sqrt( u^2 +- a^2).

u^2 = 375.67 t^2 so u = `sqrt(375.67) * t = 19.382 t approx.

Similarly a = `sqrt(715.86) = 26.755 approx..

integral(sqrt(375.6 t^2 - 715.86), t from 8 to 16) will be about 1850. Dividing this by the length 8 of the interval gives us the average value, which is about 1850 / 8 = 230.

** THE FUNCTION IS CLOSE TO THE LINEAR FUNCTION 19.4 t. The 715.86 doesn't have much effect when t is 8 or greater so the function is fairly close to P = 19 t. This approximation is linear so its average value will occur at the midpoint t = 12 of the interval. At t = 12 we have P = 19 * 12 = 230, approx.

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Self-critique (if necessary):

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