integrate – Solve a definite integral with boundary constraints

I have the following double definite integral.

$ int _0 ^ { frac {d} {1-r}} int _ { frac {d + r x-x} {r}} ^ s [(1-r) x+r y -d]$ dydx

with the constraints: $ 0 leq d leq $ 1, $ 1 leq s leq $ 4, $ 0 leq r leq $ 1, $ 0 leq x leq s $, $ 0 leq y leq s $.

My code to solve this integration is

To integrate[(1 - r)*x + r*y - d, {x, 0, d/(1 - r)}, {y, (d + r*x - x)/r,s}]

And I had

(d 3 - 3 d 2 + 3 + 2) - (6 - 6)

But I think it's not correct because the variables $ x $ and $ y $ are linked by $ 0 and $ s $ and so I think I need additional constraints on the boundaries, that is to say

$ frac {d} {1-r} leqs $ and $ 0 leq frac {d + rx-x} {r} leq $.

And I wonder how to add these two constraints to solve the integration above. Can someone help you please? Thank you!

integration – Legendre Polynomial Integral on half of the space

I need to calculate the next integral
$$
I_ {n, m}: = int_0 ^ 1 P_n (x) P_m (x) ; mathrm {d} x
$$

or $ P_n $ is the Legendre polynomial.

For an equal sum $ n + m = $ 2l it's easy to show that
$$
I_ {n, m} = frac {1} {2} int _ {- 1} ^ 1 P_n (x) P_m (x) ; mathrm {d} x
= delta_ {n, m} frac {1} {2n + 1} ,.
$$

A length calculation occurs for a odd sum $ n + m = 2l + 1 $. Has someone finished the calculation?

Is there a way to solve this integral?

I saw an answer (in terms of BesselY and StruveH functions) to this integral:

To integrate[E^(-R1/Rd)R1/Sqrt[R1^2+z^2], {R1,0, Infinity}, {f, 0,2Pi}]

However, it seems that the Mathematica can not do this integration. Do you have an idea?

Is there a simple polyhedral characterization of these integral points?

Given $ n in mathbb Z _ {> 0} $ consider all of the $ n $-tuples $$ (a_1, dots, a_n) in mathbb Z _ { geq0} ^ n $$ on the following simple conditions

  1. $ 0 leq a_i leq 2 ^ {2 ^ n} -1 $

  2. Yes $ a_i = b_ {i, 2 ^ {n} -1} 2 ^ {2 ^ {n} -1} + points + b_ {i, 0} $ were $ b_ {i, j} in {0,1 } $ each $ {0,1 } ^ n $ tuple occurs as $ (b_ {1, j}, dots, b_ {n, j}) in {0,1 } ^ n $ to something unique $ j in {0, dots, 2 ^ n-1 } $.

  1. Is there a terminology for these $ n $-tuples in literature?

  2. Is there a polyhedral characterization $ mathbb R ^ n $ who captures such an integer $ n $-tuples?

Find the next integral in terms of parameters

If I have $$ int_0 ^ infty {e ^ {- au ^ 2 + bu-c on ku} on u ^ 2} from $$ or $ a, b, c, k in mathbb {R} _ + $.

Is there a way to get an explicit formula in terms of constants? All the clues are appreciated.

calculation and analysis – Evaluating a symbolic integral

I'm trying to integrate the entire suite symbolically via to integrate order:

$$ (0.09) Bigg[1 + Bigg{ int_{3t-4k-7}^{2k+7/3} (0.09) text{exp}Bigg(int_{3s-4k-7}^{2k+7/3} (0.09) text{exp}Bigg(int_{3u-4k-7}^{2k+7/3}(0.09)dxi Bigg)duBigg)dsBigg} Bigg]$$

but it gives a result like this:

$$ 0.09 (1 + 0.06 (-1.56 + 1.23 text {ExpIntegralEi})[1482e^{4.82k-2.43t}]$$

and I can not evaluate this expression further, even if I substitute integers with $ k.

How can I continue further from this expression?

Edit: Mathematica code

integral1 = !  (
 * SubsuperscriptBox[([Integral]),  (3 u - 4k - 7 ),  (2k +
7/3 )

 ( ((0,09) ) [DifferentialD][Xi]) )

integral2 = !  (
 * SubsuperscriptBox[([Integral]),  (3s - 4k - 7 ),  (2k +
7/3 )

 ( ((0,09) ) Exp[integral1] [DifferentialD]u ) )

integral3 = !  (
 * SubsuperscriptBox[([Integral]),  (3 t - 4 k - 7 ),  (2 k +
7/3 )

 ( ((0,09) ) Exp[integral2] [DifferentialD]s ) )
`` `

conditional – Draw an integral where the integral depends on the parameter values ​​(`If`)

With my function $ f (x) $ J & # 39; would Plot3D

$ int_a ^ b f (x) dx $

with the parameter values $ a in [0,1]$ and $ b in [1,3]$.
The difficult part has to do with my integrand, which is as follows

$ f (x) = x $ if $ a leq b leq $ 2b,

$ f (x) = x ^ 2 $ if $ b> $ 2a.

My Mathematica code is:

f = !  ( * SubsuperscriptBox[([Integral]), (a B)](Yes[a <= b <= 2 a, x, x^2] [DifferentialD]X ) ); Flatten[Table[{a, b, f}, {a, 0, 1, .1}, {b, 1, 3, .1}], 1]

When I run this, I get so many combinations of $ (a, b) $ with the associated integral value Indefinite. I wonder if there is something wrong with my code.
Thank you!

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.

abstract algebra – If $ p (X), q (X) in R[X]$ are primitive and $ R $ is an integral domain, so $ ap (X) = bq (X) $ involve $ a $ and $ b $ are associated.

Proposal: leave $ a, b in R $. Yes $ p (X), q (X) in R[X]$ are primitive and $ R $ is an integral domain, so $ ap (X) = bq (X) $ imply $ a $ and $ b $ are associated; so are it $ p (X) $ and $ q (X) $.

Evidence:

Let
begin {align}
p (X) & = p_mX ^ m + p_ {m-1} X ^ {m-1} + cdots + p_1X + p_0 \
q (X) & = q_nX ^ n + q_ {n-1} X ^ {n-1} + cdots + q_1X + q_o
end {align}

By definition of a primitive,
begin {align}
forall r, overline {r} in R ((r mid p_0, p_1, cdots, p_m quad land quad overline {r} mid q_0, q_1, cdots, q_n) Leftrightarrow ( text {$ r $ and $ overline {r} $ are units of $ R $.})).
end {align}

assume $ ap (X) = bq (X) $then $ m = n $ and $ forall i = 0,1, cdots, n (ap_i = bq_i) $. It follows, $ a mid bq_0, bq_1, cdots, bq_n $ and $ b mid ap_0, ap_1, cdots, ap_n $. Note that $ a $ and $ b $ are either first or not. If one of them is first, then, by Lemma 3, $ b nmid p_0, p_1, cdots, p_n $. It follows, $ a = bu $ and since prime numbers are irreducible, $ u in R $ is a unit. Therefore, $ a $ and $ b $ are associated. Also, by substituting $ a $ with $ bu $ and cancellation of $ b $, $ p (X) $ and $ q (X) $ are associated.

I do not know how to proceed when neither $ a $ or $ b $ are first. 🙁

Integral exponential + limit

Let $ f: mathbb {R} – {- 5 } -> mathbb {R} $, $ f (x) = (x-1) e ^ {- (1 / (x + 5))} $.
I have to calculate $ lim _ {(n -> infty)} = n ^ 2 int_ {0} ^ {1} x ^ nf (x) dx $.

I've tried using the integration by parts, but I'm still stuck.