## How to calculate the integral \$ int _ {- 10} ^ {10} frac {3 ^ x} {3 ^ { lfloor x rfloor}} dx \$?

I have to calculate this integral:
$$int _ {- 10} ^ {10} frac {3 ^ x} {3 ^ { lfloor x rfloor}} dx$$

I know this function to know. $$3 ^ {x- lfloor x rfloor}$$ is periodic with period $$T = 1$$ so I've rewritten the integral as $$20 int_ {0} ^ {1} frac {3 ^ x} {3 ^ { lfloor x rfloor}} dx$$

But the problem is that I can not understand how to calculate the final integral.

Any help is appreciated.

## How to evaluate \$ int_ {0} ^ { infty} ln ^ 2 (x) ln (1 + x) ln ^ 2 left (1 + frac {1} {x} right) frac { dx} {x} \$

$$I = int_ {0} ^ { infty} ln ^ 2 (x) ln (1 + x) ln ^ 2 left (1 + frac {1} {x} right) frac { mathrm dx} {x}$$

$$ln (1 + x) = x- frac {x ^ 2} {2} + frac {x ^ 3} {3} – cdots$$

$$int_ {0} ^ { infty} left (1- frac {x} {2} + frac {x ^ 2} {3} + cdots right) left[ln(x)lnleft(1+frac{1}{x}right)right]^ 2 mathrm dx$$

This integral takes the form of $$J = int_ {0} ^ { infty} x ^ n left[ln(x)lnleft(1+frac{1}{x}right)right]^ 2 mathrm dx, n ge0$$

$$u = left[ln(x)lnleft(1+frac{1}{x}right)right]^ 2$$

$$u ^ {# 1} = frac {2 ln (x) ln (1 + 1 / x) left[(1+x)ln(1+1/x)-ln(x)right]} {x (1 + x)}$$

$$v = frac {x ^ {n + 1}} {n + 1}$$

$$J = frac {x ^ {n + 1}} {n + 1} left[ln(x)lnleft(1+frac{1}{x}right)right]^ 2- frac {2} {n + 1} int_ {0} ^ { infty} x ^ {n + 1} cdot frac { ln (x) ln (1 + 1 / x) left[(1+x)ln(1+1/x)-ln(x)right]} {x (1 + x)} mathrm dx$$

Wow … it gets too hard, I'm totally lost, no help.

## inequality – Given two increasing continuous functions \$ f, g \$ prove that \$ (ba) int ^ b_a f (x) g (x) dx> int ^ b_a f (x) dx int ^ b_a g (x) \$ dx

This is the integral inequality of Chebyshev. The following proof is
http://imar.ro/journals/Mathematical_Reports/Pdfs/2010/2/Niculescu.pdf
(Theorem 3):

If \$ f \$ and \$ g \$ both increase (or decrease both) then
\$\$ tag {*}
0 le bigl (f (x) – f (y) bigr) cdot bigl (g (x) – g (y) bigr)
\$\$
for all \$ x, y in [a, b]\$. It follows that
\$\$
0 the int_a ^ b int_a ^ b bigl (f (x) – f (y) bigr) cdot bigl (g (x) – g (y) bigr) , dx dy \
= 2 (ba) int_a ^ bf (x) g (x) , dx – 2 left ( int_a ^ bf (x) , dx right) left ( int_a ^ bg (x) , dx right) ,.
\$\$

If \$ f \$ goes up and \$ g \$ goes down (or vice versa), then the
the inverse inequality is valid.

Yes equality then holds the equals worth in \$ (*) \$ for all \$ x, y
in [a, b]\$ (since \$ f \$ and \$ g \$ are supposed to be continuous).
In particular
\$\$
0 = bigl (f (a) – f (b) bigr) cdot bigl (g (a) – g (b) bigr)
\$\$
which means that (at least one of) \$ f \$ or \$ g \$ is constant.

## calculation – \$ left | frac {d ^ n} {dx ^ n} (x ^ 2-1) ^ n right | sqrt frac2 pi cdot frac {2 ^ nn!} { sqrt n} cdot frac1 { sqrt[4]{1-x ^ 2}} \$, a binding for the Legendre polynomial

Question
CA watch
I)$$left | frac {d ^ n} {dx ^ n} (x ^ 2-1) ^ n right | sqrt frac2 pi cdot frac {2 ^ nn!} { sqrt n} cdot frac1 { sqrt[4]{1-x ^ 2}},$$ or equivalent $$left | P_n (x) right | sqrt frac2 { pi n} cdot frac1 { sqrt[4]{1-x ^ 2}} text {(where P is the Legendre polynomial)},$$ when $$x in (-1,1)$$ and $$n in mathbb Z ^ +$$.
ii) The constant term $$sqrt frac2 pi$$ in (i) is the best.

I have successfully proved the statement (ii) using the asymptotic development of $$P_n$$ given by Wikipedia:
$$P_n ( cos theta) = sqrt frac2 { pi n sin theta} cos ((n + 1/2) theta- pi / 4) + O (n ^ {- 1} ).$$ Taking supremum limit,
$$limsup_ {n to infty} | P_n ( cos theta) | sqrt n le sqrt frac2 pi sqrt[4]{1- cos ^ 2 theta}.$$
Replacing $$cos theta = x$$we can see that (ii) is a weaker result of the above limit result. However, I do not know how to deduce the explicit limit of $$P_n (x)$$.

## calcul – Fresnel integral improper \$ int_0 ^ infty sin (x ^ 2) dx \$ question

I found an example of an inappropriate Fresnel Integral $$int_0 ^ infty sin (x ^ 2) dx$$ calculation. He uses several substitutions. The next substitution is not clear to me.

I did not understand how to get the right side of the left side. What subtitling is done here?

$$int_0 ^ infty frac {v ^ 2} {1 + v ^ 4} dv = frac {1} {2} int_0 ^ infty frac {1 + u ^ 2} {1 + u ^ 4} of.$$

## How can I study the convergence of the incorrect integral \$ int_ {2} ^ {+ infty} frac {dx} {x ln ^ 2x} \$?

I have tried to apply comparison criteria and limit criteria (by reducing it to a form of $$p$$-series) in vain.

Rather than discovering this particular solution, I would like to know how to approach even this kind of exercises.

## to prove \$ dx ^ 2 + dy ^ 2 + dz ^ 2 = dr ^ 2 + r ^ 2 d ( theta) ^ 2 + r ^ 2 (sin theta) ^ 2 d phi ^ 2 \$

$$dx ^ 2 + dy ^ 2 + dz ^ 2 = dr ^ 2 + r ^ 2 d ( theta) ^ 2 + r ^ 2 (sin theta) ^ 2 d phi ^ 2$$

given $$x = rsin theta cos phi, y = rsin theta sin phi , z = r cos theta$$

How can I do this?

## computation – How do you integrate \$ int_ {0} ^ { infty} frac {a cos {(cx)}} {a ^ 2 + x ^ 2} dx \$?

This integral haunts me for some time, because it escapes all the methods of integration that I could find (substitution in u, integration by parts, trigonometric substitution and same method of Feynman). I realize that this is not elementary, but I can not find how to find the definite integral. I know you'll have to use Feynman's method, but I'm lost.

To be clear, I want to know How to integrate it, not what is the value.

## dx

BlackHatKings: Proxies and VPN Section
Posted by: MervinROX
Post time: June 8, 2019 at 05:12.

## Integration – Verification: \$ left | int_0 ^ 1 f (x) dx right | leq frac {1} {2} int_0 ^ 1 | f (x) | dx. \$

Let $$f (x)$$ to be a function with $$f (x)$$ continue on $$[0,1]$$. $$f (0) + f (1) = 0$$. Prove $$left | int_0 ^ 1 f (x) dx right | leq frac {1} {2} int_0 ^ 1 | f (x) | dx.$$

Let $$x-1/2 = t$$,then $$x = t + 1/2$$,$$dx = dt$$. So
begin {align *} left | int_0 ^ 1 f (x) dx right | & = left | int _ {- frac {1} {2}} ^ {~~ frac {1} {2}} f left t + frac {1} {2} right) dt right | \ & = left | left[tfleft(t+frac{1}{2}right)right]_ {- frac {1} {2}} ^ {~~ frac {1} {2}} – int _ {- frac {1} {2}} ^ {~~ frac {1} { 2}} tf & # 39; left (t + frac {1} {2} right) dt right | ~~~ & textit {integrating by parts} \ & = left | 0- int_ {0} ^ {1} left (x- frac {1} {2} right) f left (x right) dx right | ~~~ & textit {substituting t + 1/2 = x } \ & leq int_0 ^ 1 left | left (x- frac {1} {2} right) f left (x right) right | dx \ & leq left | xi- frac {1} {2} right | int_0 ^ 1 | f (x) | dx ~~~ & textit {applying the first MVT to the integral} \ & leq frac {1} {2} int_0 ^ 1 | f (x) | dx end {align *}
what is desired.