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\leqcos x \leq1

-1\leq(cos x)/(x +exp(x))\leq1/(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} =&gt; dx=2tdt, <br /> t=\sqrt x x=0 =&gt;t=0, when, x=b =&gt; 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!