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Calculus/Calc 2 evaluating improper integrals
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Question
evaluate the improper integrals or show that it diverges.
1. integral from 0 to 9 of ((e to the sq. root x) divided by sq. root x).
2. integral from 1 to infinity of (x+1)/x squared
Let's start with section 2:
∞
∫ (x+1)/x dx
1
We will use the rule :
If f(x)>g(x) in open interval [1,∞) &
∞
∫ f(x) dx Diverge, Then
1
∞
∫ g(x) dx Diverge as as well.
1
We know that (x+1)/x = 1+1/x , & we know also that 1+1/x > 1/x.
∞
∫ 1/x dx = Lim Ln(t) = ∞
1 t->∞
Thus :
∞
∫ (x+1)/x dx -> ∞
1
---------
As for section 1:
9
∫ [e^(√x)]/(√x)
0
We will evaluate the integral using Trapezoid Rule :
∆x=(9-0)/n where n=4.
9
∫ f(x) dx = [∆x/2]*[f(xo)+2f(x1)+2f(x2)+..+2f(xn-1)+f(xn)]
0
Thus :
x0=9/4 x1=18/4 x2=27/4 x3=9
9
∫ [e^(√x)]/(√x) = [9/8]*[e^(√(9/4)/√(9/4) + e^(√(18/4)/√(18/4) +
0
e^(√(27/4)/√(27/4) + e^(√9)/√9 .
I'll leave it for you to complete the calculations.
Alon.
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Answers by Expert:
Alon Mandes
Expertise
Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .
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3. 2 years of experience as a math teacher in college.
4. I give free homework help for high school students in
Mathematics & Physics.
5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" ,
"Complex Functions".
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M.A in Mathematics & Bs.c in Electronics.
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