Integral Convergence Homework Solutions

[Mattofix]

Homework Statement

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

ii)
$$\int$$$$^{\infty}_{}$$ (x + $$\sqrt{x})^{-1}$$dx
$$_{ 1}$$

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

The Attempt at a Solution

i) x +exp(x) $$\geq$$ 1

-1$$\leq$$cos x $$\leq$$1

-1$$\leq$$(cos x)/(x +exp(x))$$\leq$$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.

 

[sutupidmath]

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
$$\int_0^{\infty} \frac{cos x}{x+e^{x}}dx$$ converges

 

[sutupidmath]

For the other two i believe you can find the antiderivatives of those functions, and see whether they converge or not!


$$\int_0^{\infty}\frac{dx}{x+\sqrt x}=\lim_{b\rightarrow\infty} \int_0^{b}\frac{dx}{x+\sqrt x}$$, now let $$x=t^{2} => dx=2tdt, <br /> t=\sqrt x$$ $$x=0 =>t=0, when, x=b => t=\sqrt b$$
$$\lim_{b\rightarrow\infty} \int_0^{\sqrt b}\frac{2tdt}{t^{2}+t}=2\lim_{b\rightarrow\infty}\int_0^{\sqrt b}\frac{dt}{1+t}$$,

 

[sutupidmath]

I think also for the iii) you will be able to find an antiderivative in terms of an el. function!