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}]
``````

``````(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.