Questions about the evaluation of specific definite integrals.

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2
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1answer
93 views

Show that $\int^{\infty}_0 \frac{\sin^4 x}{x^4}=\frac{\pi} 3$.

Show that $$\int^{\infty}_0 \left( \frac{\sin x}{x}\right)^4=\frac{\pi} 3$$ Although I know the integral with the index is $1$ and $2$, I have no idea on this one. Please help.
0
votes
2answers
17 views

Probability that the call will be answered at time $t$ is given by $f(t)$. Find the median waiting time for the call.

$$f(t) = \begin{cases} 0 & \text{if $t < 0$ } \\ 0.2e^{-t/5} & \text{if $t\geq 0$} \end{cases}$$. $ $ Find the median waiting time for the call. $ $ I cannot understand ...
3
votes
0answers
33 views

A probabilistic integral $\int_{-\infty}^{\infty}e^{-x^2/2\sigma^2}\arcsin\left(1-2\left|\lfloor x\rceil-x\right|\right)\,dx$

In my probabilistic studies, a tough integral appeared. Note that $\lfloor x\rceil$ is not the floor function; it is the nearest integer function. Up to some constants, it appears in a Buffon-like ...
1
vote
2answers
87 views

Integration of $x\cos(x)/(5+2\cos^2 x)$ on the interval from $0$ to $2\pi$

Compute the integral $$\int_{0}^{2\pi}\frac{x\cos(x)}{5+2\cos^2(x)}dx$$ My Try: I substitute $$\cos(x)=u$$ but it did not help. Please help me to solve this.Thanks
0
votes
1answer
23 views

Use the form of the definition of the integral to evaluate the integral (4-2x)

Use the form of the definition of the integral to evaluate the integral. $$a=2, b=5, (4-2x)dx$$ $$\lim_{n \to \infty} \sum_{i=1}^{n}(4-2x)\Delta x$$ $$\Delta x = \frac{3}{n}$$ $$x_i=2+i\frac{3}n$$ ...
0
votes
0answers
15 views

Proving Taylor Series Estimation of Integrals

Given that $f, g, h$ are all continuos on $x \in [a,b]$, $g(x)$ approximates $f(x)$ with an error of at most $h(x)$ Meaning $|f(x) - g(x)| \le h(x)$ Have to prove $|\int_a^bf(x)\,dx - ...
4
votes
0answers
63 views

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
3
votes
1answer
64 views

Evaluating $\int^{\frac{\pi}{2}}_0 \sin^n x ~\mathrm{d}x$

I'm trying to find the general formula for the following: $$I_n = \int^{\frac{\pi}{2}}_0 \sin^n x ~\mathrm{d}x$$ I remember doing it a while back but for the life of me, I can't remember right now. ...
5
votes
4answers
337 views

Arc Length of a Curve

Let $f:[a,b]\to \mathbb{R}$ be a continuous function, how can you prove (not in the geometric way): $$ \sqrt{\left(f(b)-f(a)\right)^2+\left(b-a\right)^2}\le\int_a^b \sqrt{1+f'(x)^2}dx $$
4
votes
3answers
89 views

Integral of $\int_{0}^{y} \exp\left(\, -\alpha x\,\right)\, x \sqrt{1-x^2}{\rm d}x$

Does the following integral have a closed form solution? $$ \int_{0}^{y} \exp\left(\, -\alpha x\,\right)\, x \sqrt{1-x^2}{\rm d}x $$ Or is there an approximation which works for large $\alpha$?
0
votes
0answers
14 views

change of limits of integration, absolute pressure to gauge pressure

I have this relation $(P_s(x,t)+b)$, where $P_s(x,t)$ is absolute pressure as a function of position, $x$, and time, $t$, and $b$ is some constant. Recall that absolute pressure is equal to gauge ...
3
votes
2answers
147 views

Integral of exponential with $x(1-x)$ term

Does the following integral have a closed form solution? $$ \int_{0}^{y} \exp\left(\,\sqrt{\,x(1-x)\,}\,\right)\,{\rm d}x $$ Or must I settle with an approximation? Edit: Actual form of integral ...
6
votes
0answers
125 views
2
votes
1answer
53 views

Intuition concerning Riemann Sums

I have just started learning integrals, and I want to know the following: In the definition of a riemann integral, it states that the interval that the integral is to be evaluated, is partitioned ...
2
votes
3answers
85 views

Evaluate $\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} $

$$\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} dx$$ My approach is to calc $$\int_{1}^{X} \frac{\ln{(2x-1)}}{x^2} dx$$ and then take the limit for the answer when $X \rightarrow \infty$ However, I must ...
0
votes
3answers
33 views

Integral of a function with two parts (piecewise defined)

The function has 2 parts: $$f(x) = \begin{cases} -\sin x & x \le 0 \\ 2x & x > 0\end{cases}$$ I need to calculate the integral between $-\pi$ and $2$. So is the answer is an integral ...
1
vote
0answers
79 views

Riemann implies Lebesgue integrablility on $\mathbb{R}^n$, prove $f(x)$ continuous at x where $g(x)=G(x)$

Let $f:[a_1,b_1]\times \cdots \times[a_n,b_n] \rightarrow \mathbb{R}$ be Riemann integrable. Prove that is $f$ Lebesgue integrable. Proof: $$Q:= [a_1,b_1]\times \cdots \times [a_n,b_n].$$ For simple ...
0
votes
2answers
20 views

Average Value - Graphs

long method: Determine an equation for each and solve using average value formula alternative methods? How could you prove the average value to be C over an interval [a,b] if you are given a ...
-1
votes
0answers
26 views

Darboux integrable, $f$ continuous at x where g(x)=G(x) [duplicate]

$f:[a_1,b_1]x[a_2,b_2]\rightarrow \mathbb{R}$ that is Riemann integrable, and let $g(x),G(x)$ functions with property $g(x)\leq f(x) \leq G(x)$, g=G a.e.! G(x), g(x) are obtain from proof Riemann int ...
1
vote
0answers
26 views

Integral/infinite sum related to Bessels which pop up in optical coherence theory

In propagating partially coherent optical fields, the following integral pops up: $I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta$, where a and b are real numbers. If we consider ...
0
votes
2answers
40 views

How to use trigonometric substitution to compute this definite integral?

I have searched for a similar question on stack exchange but could not find one. The definite integral: $\large\int_0^1 \frac{x^4}{\sqrt{25-x^2}}$ I realize that I need to use $x = 5\sinθ$ in the ...
0
votes
1answer
47 views

Symbolic Integration involving hypergeometric functions

What's the best way to symbolically evaluate this integral? $$\frac{1}{\hbar}\int_{-\infty}^\infty e^{iux/\hbar}\Psi^{*}_n(p-u/2)\Psi_n(p+u/2)\,du$$ where: $$\Psi_n(p)=\frac{1}{(1+\alpha ...
1
vote
1answer
46 views

Determining the infinite limit of a Riemann' sum

I need to evaluate the following $f(x)=x^2 + 2x - 5$, on $[1,4]$, by using the riemans sum and limiting it to infinity. I have set up everything $\Delta x=\frac{3}{n}$. $x_i=1+\frac{3i}{n}$, I would ...
2
votes
1answer
90 views

Examples of pairs of difficult integrals

I’m looking for pairs of difficult definite integrals that are linked algebraically on a certain field without known change of variable or integration by parts from one integral to the other. Here a ...
2
votes
1answer
42 views

Evaluation of the definite integral: $\int_0^{\pi/2}x\tan(x)^pdx$

The integral: $$J=\int_0^{\pi/2}x\tan(x)^pdx$$ has the solution: $$J=\dfrac{\pi}{4\sin\left(\frac{\pi}{2}p\right)}\left[\Psi\left(\dfrac{1}{2}\right)-\Psi\left(\dfrac{1-p}{2}\right)\right]$$ where the ...
3
votes
3answers
200 views

Prove $\int_{\mathbb{R^{+}}} \frac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\frac{\pi}{32} (3\pi-4)^2$

How do you arrive at the result $$I=\displaystyle\int_{\mathbb{R^{+}}} \dfrac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\dfrac{\pi}{32} (3\pi-4)^2\ ?$$ Wolfram Alpha agrees numerically. I tried ...
2
votes
2answers
91 views

How to do integral $\int_0^{\infty} e^{-x^2-ax^4}\ dx , \ \text{ for $a>0$}$

I was told by this OP, $$\int_{0}^{\infty} e^{\large-x^n} \,dx =\Gamma \left(\frac{n+1}{n}\right), \qquad\text{ for $n>1$}.$$ This is via the variable change $t=x^n$: $$\int_{0}^{\infty} ...
0
votes
2answers
108 views

Improper integral of $\frac{\ln x}x$

Find $$\int_e^{\infty}\frac{\ln x}{x}\ dx$$ $A.\ \dfrac12$ $B.\ \dfrac{e^2}{2}$ $C.\ \dfrac{\ln(2e)}{2}$ $D.$ DNE (Does not exist) I tried doing this and this is where I've gone so far: $$\lim ...
0
votes
1answer
34 views

$f(r) \leq \int_r^{r+1} f(t)dt$

Suppose $f:[0,\infty)\to [0,\infty)$ is continuous (uniformly, if you want) and that $\int_0^{\infty} f(t)~\mathrm{d}t < \infty$. Is the following true? $$ f(r) \leq \int_r^{r+1} f(t)~\mathrm{d}t ...
3
votes
1answer
21 views

What is the value of $a$ so that this condition holds?

Let $f(x) \colon= x-x^2$, $g(x) \colon= ax$. Determine the value of $a$ so that the region above the graph of $g$ and below the graph of $f$ has area equal to $9/2$. Here $f(x) - g(x) = (1-a)x - x^2 ...
1
vote
2answers
56 views

How are these two integrals related?

How to express the integral $$\int_{-2}^{2} (x-3) \sqrt{4-x^2} \ dx $$ in terms of the integral $$ \int_{-1}^{1} \sqrt{1-x^2} \ dx?$$ I know that the latter integral is equal to $\pi / 2$. We can't ...
2
votes
2answers
27 views

Double integral where limits are the first quadrant

Evaluate the integral $$\iint\limits_D \frac{1}{(x+y+1)^3} \, dA$$ where $D$ is the first quadrant. In this case, what would the limits of integration be? I'm having trouble moving to polar ...
1
vote
0answers
20 views

Express the limit as a definite integral on the given interval [Difficulty Interpreting Question]

I understand that they want the integral as an answer, but how would I go about getting this. Sorry this is all just very new to me and i just want to grasp the concepts :)
1
vote
3answers
112 views

Evaluate $\int_0^\infty\frac{dl}{(r^2+l^2)^{\frac32}}$

How to evaluate the following integral $$\int_0^\infty\frac{dl}{(r^2+l^2)^{\large\frac32}}$$ The solution is supposed to look like this, unfortunately I can't derive it. $$ ...
3
votes
3answers
49 views

The average value of the function $y=\tan(2x)$ over the interval $[0,\frac{\pi}{8}]$

I was given the following question in a technology free exam. How would one go about solving this without the use of a calculator? NB. I am currently in my last year of high school so please don't ...
5
votes
2answers
114 views

How to do integral $\int_0^T \frac{1}{t\sqrt{t(T-t)}}e^{-\frac{b^2}{2t}}dt$ and $\int_0^T \frac{1}{\sqrt{t(T-t)}}e^{-\frac{b^2}{2t}}dt$?

I met these two integrals but don't know how to do them: $$I_1 = \int_0^T \frac{1}{t\sqrt{t(T-t)}}e^{-\frac{b^2}{2t}}dt$$ $$I_2 = \int_0^T \frac{1}{\sqrt{t(T-t)}}e^{-\frac{b^2}{2t}}dt$$ where ...
0
votes
1answer
17 views

Average and Median Distance integral [on hold]

I know how to solve part A. I am having problems solving parts B & C.
8
votes
2answers
152 views

A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$?

The following integrals are classic, initiated by L. Euler. \begin{align} \displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} ...
1
vote
1answer
58 views

Finding a mistake in the computation of a double integral in polar coordinates

I have to find $P\left(4\left(x-45\right)^2+100\left(y-20\right)^2\leq 2 \right) $ $f(x)$ and $f(y)$ are given, which I will use in my solution below . ...
10
votes
0answers
120 views
2
votes
1answer
42 views

Computing double integral

Find $$\iint\limits_D \sqrt{(x-10)^2+y^2}\hspace{1mm}dA$$ where $\{(x, y)\in D \mid x^2+y^2\leq 10^2\}$. I am not sure how to start, every way I have tried so far, ends up into something ugly. All ...
1
vote
5answers
101 views

Definite integral $\int_{-\pi/2}^{\pi/2}\cos^{2}\left(\theta\right)\,{\rm d}{\theta} $

Please help me to evaluate definite integral $$\int_{^{-\pi}/_2}^{^\pi/_2}\cos^{2}\left(\theta\right)\,{\rm d}{\theta}$$ Also there was a hint: Use the double angle formula ...
1
vote
3answers
115 views

Value of the integral $\int_{\mathbb{R}} \frac{x\sin {(\pi x)}}{(1+x^2)^2}$

How do we evaluate the integral $$I=\displaystyle\int_{\mathbb{R}} \dfrac{x\sin {(\pi x)}}{(1+x^2)^2}$$ I have wasted so much time on this integral, tried many substitutions $(x^2=t, \ \pi x^2=t)$. ...
1
vote
3answers
88 views

Evaluate $\int_{1}^{e}\frac{u}{u^3+2u^2-1}du.$

I'm trying to solve $$\int_{1}^{e}\frac{u}{u^3+2u^2-1}du.$$ My first approach was to factorise and then do a partial integration. However the factorisation ...
6
votes
0answers
106 views

An incorrect answer for an integral

As the authors pointed out in this paper (p. 2), the following evaluation which was in Gradshteyn and Ryzhik (sixth edition) is incorrect (and has been removed). $$ ...
3
votes
2answers
238 views

What is the value of this double integral?

Let $C$ be the subset of the plane given by $$ C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x^2 + y^2 \leq 1 \}.$$ Then what is the value of the double integral $$ \int_{C} \int (x^2 + y^2) ...
0
votes
2answers
61 views

How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}. $$ Then how to evaluate the double integral $$ \int_C \int (x^2+ y^2) dx dy? $$ My ...
0
votes
2answers
40 views

What is the area bounded by these curves?

Let $f(x) \colon = x^2$, $g(x) \colon= x+1$. Then what is the area bounded by the graphs of $f$ and $g$ between the vertical lines $x= -1$ and $x= (1+\sqrt{5})/2$? My effort: Since $$ f(x) - g(x) ...
0
votes
2answers
45 views

Integration of piecewise defined function: $ f(x)=0$ for $x<1$ and $f(x)=1$ for $x\geq1$

I think I am confusing myself too much on this. Let $ f(x)=0$ for $x<1$, and $f(x)=1$ for $x\geq1$. What is $\int_0^1f(x)\,dx$? I am worried because $f$ is discontinuous at $1$. Does that make ...
10
votes
1answer
198 views
+100

Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$

I discovered the following conjecture numerically, but have not been able to prove it yet: $$_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47.\tag1$$ The equality holds ...