Integral Convergence Homework Solutions

In summary, i) the function x+exp(x) is greater than 1, ii) the function x+\sqrt{x} is less than 1, and iii) the function f(x)dx<g(x) if and only if integ(from a to x)f(x)dx<M.
  • #1
Mattofix
138
0

Homework Statement



i)
[tex]\int[/tex][tex]^{\infty}_{}[/tex] (cos x)/(x +exp(x)) dx
[tex]_{ 0}[/tex]

ii)
[tex]\int[/tex][tex]^{\infty}_{}[/tex] (x + [tex]\sqrt{x})^{-1}[/tex]dx
[tex]_{ 1}[/tex]

iii)
[tex]\int[/tex][tex]^{\infty}_{}[/tex] (1 + x[tex]^{3}[/tex])[tex]^{-1/2}[/tex] dx
[tex]_{ 1}[/tex]

The Attempt at a Solution

i) x +exp(x) [tex]\geq[/tex] 1

-1[tex]\leq[/tex]cos x [tex]\leq[/tex]1

-1[tex]\leq[/tex](cos x)/(x +exp(x))[tex]\leq[/tex]1/(x +exp(x))

then do i have to compare something to something knowing for certain that something convereges?

For ii) + iii) please can someone nudge me in the right direction.
 
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  • #2
for the first one use the fact that cosx<1 for x from zero to infinity, also use the fact that 1/(x+exp(x))<exp(-x), then try to show that exp(-x) converges, so there is a theorem i guess, i am not sure how exactly it goes but i think it say that if

f(x)<g(x), then also

integ (from a to x)f(x)dx<integ(from a to x) g(x), then if the right hand sided integral converges say to a nr M, then it means that integ (from a to x)f(x)dx<M, so it means that this function is upper bounded so it also must have a precise upper bound, hence the limit also must exist as x-->infinity, which actually tells us that the integral
[tex]\int_0^{\infty} \frac{cos x}{x+e^{x}}dx[/tex] converges
 
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  • #3
For the other two i believe you can find the antiderivatives of those functions, and see whether they converge or not!


[tex]\int_0^{\infty}\frac{dx}{x+\sqrt x}=\lim_{b\rightarrow\infty} \int_0^{b}\frac{dx}{x+\sqrt x}[/tex], now let [tex]x=t^{2} => dx=2tdt,
t=\sqrt x[/tex] [tex]x=0 =>t=0, when, x=b => t=\sqrt b[/tex]
[tex]\lim_{b\rightarrow\infty} \int_0^{\sqrt b}\frac{2tdt}{t^{2}+t}=2\lim_{b\rightarrow\infty}\int_0^{\sqrt b}\frac{dt}{1+t}[/tex],
 
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  • #4
I think also for the iii) you will be able to find an antiderivative in terms of an el. function!
 

Related to Integral Convergence Homework Solutions

What is "Integral Convergence Homework Solutions"?

Integral Convergence Homework Solutions is a set of solutions to homework problems related to integral convergence, a topic in calculus. It is designed to help students better understand and solve problems related to integral convergence.

Why is integral convergence important?

Integral convergence is important because it allows us to determine the convergence or divergence of an infinite series or improper integral. This is useful in many real-world applications, such as in physics, engineering, and economics.

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You can use Integral Convergence Homework Solutions as a study tool to check your solutions and learn from the explanations provided. By comparing your own work to the solutions, you can identify any mistakes and improve your understanding of the concepts.

Are the solutions provided in Integral Convergence Homework Solutions accurate?

Yes, the solutions provided in Integral Convergence Homework Solutions are accurate and have been thoroughly checked by experienced mathematicians. However, it is always beneficial to double check your own work and understanding.

Can I use Integral Convergence Homework Solutions to cheat on my homework?

No, Integral Convergence Homework Solutions should not be used to cheat on homework. It is intended to be used as a study guide and reference tool to help you better understand the material, not to replace your own efforts and learning.

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