Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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3
votes
3answers
77 views

How to integrate $\int_{-\pi/4}^{\pi/4} (\cos(t) + \sqrt{1 + t^2}\cos^3(t)\sin^3(t))\;\mathrm{d}t$. [duplicate]

How do I integrate $$\int_{-\pi/4}^{\pi/4} (\cos{t} + \sqrt{1 + t^2}\cos^3{(t)}\sin^3{(t)})\;\mathrm{d}t$$ There's no substitution that immediately jumps out at me.
3
votes
0answers
39 views

Estimating the sum of a series within arbitrary certainty.

Find the sum of the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^5} = a_n$ within three decimal places. The sum is estimated by $\displaystyle a_n \approx \sum_{k=1}^{n}\frac{1}{k^5}+R(n)$ ...
35
votes
1answer
478 views

Is there an integral for $\pi^4-\frac{2143}{22}$?

In Ramanujan's Notebooks, Vol 4, p.48 (and a related one in Quarterly Journal of Mathematics, XLV, 1914) there are various approximations, including the close (by just $10^{-7}$), $$\pi^4 \approx ...
0
votes
1answer
53 views

Is there an analytic or at least a numerical solution to an eqaution of the form $\sqrt{k_1\sqrt{x}+k_2}\;\Big(k_3x+k_4\sqrt{x}+k_5\Big)+k_6=0$?

So the problem comes from cosmology and I want to solve for the unknown function $a(t)$, which is the scale factor for the universe. So I have an integral involving $a$: $$ ...
2
votes
4answers
70 views

Integral which must be solved using integration by parts

I have to solve this problem using integration by parts. I am new to integration by parts and was hoping someone can help me. $$\int\frac{x^3}{(x^2+2)^2} dx$$ Here is what I have so far: $$\int udv ...
2
votes
2answers
59 views

Calculate integral $\int_0^\infty e^{-x} (e^{-\frac a b x} - 1)^{b} dx$ for $b>0$ and any $a \in \mathbb{R}$

i am working on following task: Choose any nonzero $a \in \mathbb{R}$ so the integral converges and for a given $b > 0$ compute $\int_t^\infty e^{-x} (e^{-\frac a b x} - 1)^{b} dx$. I am looking ...
6
votes
3answers
239 views

How to integrate $\int_0^{\infty} \frac{e^{ax} - e^{bx}}{(1 + e^{ax})(1+ e^{bx})}dx$ where $a,b > 0$.

This $$\ \int_0^{\infty} \frac{e^{ax} - e^{bx}}{(1 + e^{ax})(1+ e^{bx})}dx \text{ where } a,b > 0. $$ is a problem that showed up on a GRE practice test. I believe you're supposed to use complex ...
9
votes
2answers
225 views

Calculating $\int_0^1 \frac{\operatorname{arctanh}\left(\sqrt{1-\frac{u}{2}}\right)\sqrt{\frac{2 \pi \sqrt{1-u}}{u-2}+\pi } }{u\sqrt{1-u}} \, du$

What real tools would you employ for calculating the integral below? Some useful ideas I mean. $$\int_0^1 \frac{\operatorname{arctanh}\left(\sqrt{1-\frac{u}{2}}\right)\sqrt{\frac{2 \pi ...
1
vote
2answers
34 views

Integrating the given function involving trigonometric functions

Find $\int \csc^{p/3}x \sec^{q/3} x dx $ Given - $(p,q \in I^{+} )$ and $(p+q=12)$ I tried to substitute $q = 12-p$ in the integral but didn't find anything satisfactory.
1
vote
2answers
95 views

How do I evaluate this integral :$\int \frac{\sqrt{-x^2-x+2}}{x^2}dx$?

Is there someone who can show me how to evaluate this integral: $$\int \frac{\sqrt{-x^2-x+2}}{x^2}dx.$$ I have tried many changes of variables but I haven't succeeded yet. Thank you for any help.
3
votes
2answers
115 views

Inclusions relating standard norms, in measury theory

I know that for finite measure space $(X, \mathcal A ,\mu )$ and $1\leq p< q<\infty $ , the inclusion $\mathcal L^q\subseteq \mathcal L^p\subseteq\mathcal L^1 $ holds true (applying Holder's ...
3
votes
3answers
382 views

Definite integral of the inverse of a function

We have $f:\mathbb{R}\to \mathbb{R}$ with $$f\left(x\right)=\frac{\left(-x^3+2x^2-5x+8\right)}{\left(x^2+4\right)}$$ Knowing that the function is bijective, calculate $$\int ...
1
vote
1answer
56 views

Finding the limits of the double integral $\iint dydx$

I need to find the limits of the double integral $$\int\limits_a^bf(x)dx=\iint dydx$$ such that $f(x)$ is a continues function in the domain $[a,b]$ Any hints please?
0
votes
2answers
117 views

Integrate along the vertical strip

I want to show that some integration with vertical line is bounded. function $f(\mu)$ is given by $$ f(\mu)=A^{-\sqrt{\mu}} \frac{(B_1-\sqrt\mu)}{(B_2-\sqrt\mu)(B_3+\sqrt\mu)} $$ where $f$ is defined ...
0
votes
2answers
86 views

Integrating $\int \frac{\sin^{-1}(x)}{\sqrt{1+x}}dx$ by parts

I have a question that requires me to integrate the following by parts. I have done the question but apparently my answer does not match that of wolfram alpha's. $$\int ...
5
votes
3answers
135 views

for each $\epsilon >0$ there is a $\delta >0$ such that whenever $m(A)<\delta$, $\int_A f(x)dx <\epsilon$

This is an old preliminary exam problem: Show that, for every nonnegative Lebesgue integrable function $f:[0,1]\rightarrow \mathbb{R}$ and every $\epsilon>0$ there exists a $\delta>0$ such ...
5
votes
5answers
80 views

Properties of $L^2(-1,1)$ functions

I want to show that there is no function $v \in L^2(-1,1)$ with $\int_{-1}^{1} v(x)\phi(x) dx = 2\phi(0)$ for all $\phi \in C^\infty_0(-1, 1)$ ($\phi$ is $0$ everywhere but $[-1,1] $). I know about ...
1
vote
1answer
39 views

Evaluating the line integral

The question I'm working on states: Let $C$ be the curve in $\mathbb R$ consisting of line segments from $(4,1)$ to $(4,3)$ to $(1,3)$ to $(1,1)$. Let $F(x,y) = (x+y)i + (y-1)^3 * e^{\sin(y)}j$. ...
4
votes
1answer
113 views

How to prove $\int_{0}^{\infty}\frac{e^{-\left(\sqrt{x}-a/\sqrt{x}\right)^2}}{\sqrt{x}}dx=\sqrt{\pi},\,a>0$?

We know that $$\Gamma\left(\frac{1}{2}\right)=\int_{0}^{\infty}\frac{e^{-x}}{\sqrt{x}}dx=\sqrt{\pi} $$ but it seems that, for every $a>0 $ we have ...
2
votes
2answers
49 views

How to evaluate two integrals (double and triple)? One with a conic boundary and the other with a square root boundary.

Calculate the integral: $$\iiint_V z^2 dx dy dz$$ Where $V \subset R^3$, bounded by: $y = \sqrt z$ rotated around $Oz$, and the plane $z = h$ $(h>0)$. My try: Introduce the polar coordinates ...
1
vote
1answer
28 views

Support of convolution

Assume $u \in L^1(\mathbb{R}^n)$ and $\mathrm{ess\,supp}(u) \subset U,$ where $U$ is a bounded open set. Now we compute the convolution of $u$ with a function $\eta \in C(\mathbb{R}^n)$ with ...
1
vote
1answer
46 views

Difference of two definite integrals

$F:\mathbb{R}\to \mathbb{R}$ $$F\left(x\right)=\frac{\left(x^2+ax+5\right)}{\sqrt{x^2+1}}$$ Find $a$ so that $$\int _0^2\left(F\left(x\right)\right)dx-\int _{-2}^0\left(F\left(x\right)\right)dx=2$$ I ...
6
votes
1answer
123 views

How would I complete my proof that $\int_a^bf(g(x))\,dx = \int_{g(a)}^{g(b)}f(x)\frac{d}{dx}(g^{-1}(x))\,dx$?

Around two years ago during a second semester calculus class, my professor remarked that $\int\sin(x^2)\,dx$ could not be integrated. Being a bit defiant, I tried (in vain) to prove him wrong. While ...
-1
votes
3answers
87 views
0
votes
2answers
62 views

How to solve this type of integrals?

I have specific problem for solving this type of integral : $$\int \frac{x-y}{x^2-2y^2}\, dx $$ I tried applying partial fractions, making the denominator complete square but the sum keeps on ...
0
votes
2answers
28 views

Class of differentiable functions and Lipschitz continuity

I am reading lectures notes by Dr. Yuvi Nesterov's "Introductory Lectures on Convex Programming ". On page 25, Lemma 1.2.2, to prove $f''(x) \leq L$, (where $f(x) \in C_L^{2,1}(R^n)$, $L$ is Lipschitz ...
2
votes
3answers
58 views

The limit of the integral when the set is decreasing in probability to zero.

This is an exercise problem(#2 in section 3.2) from 'A course in probability theory'. If $E(\vert X \vert ) < \infty$ and $\lim_{n \to \infty} P(A_n) = 0,$ then $\lim_{n \to \infty} \int_{A_n} X \ ...
0
votes
1answer
43 views

Is the function $\ln (u(x))$ integrable when $u$ is bounded and positive?

Consider $\Omega$ an open bounded domain in $\mathbb{R}^n$ and $ u \in L^{\infty} (\Omega)$ a positive function. My question is : the well defined function $\ln (u(x))$ is integrable? Intuitively ...
1
vote
1answer
64 views

Computing a limes via integration

Currently, I am studying for an exam (topics: real analysis, integration etc.). I came across the following exercise: Let $\lambda>-1$ and let $(a_n)$ be the sequence defined by ...
0
votes
2answers
99 views

Is my answer correct for $\int \frac{\sin(x)}{\cos^3(x)} \;\mathrm{d}x$

I said $\frac{\tan^2(x)}{2} + c$ but my book says $\frac{1}{2\cos^2(x)} + c$
11
votes
7answers
916 views

Why does the “separation of variables” method for DEs work? [duplicate]

Heyho, I am using the separation-of-variables method for quite a while now, but what was always bothering me a bit, is why is it possible to do those operations. I'll give a concrete example (source ...
2
votes
1answer
84 views

Recursive Integral.

If $\int _o^1 e^x(x-1)^ndx=16-6e$ find the value of n (n is a positive integer $n\le5$). ATTEMPT: Let $I=\int _o^1 e^x(x-1)^ndx$ By using $\int _a^b f(x)dx=\int _a^b f(a+b-x)dx$ $I=\int _o^1 ...
3
votes
2answers
46 views

Limiting case of of integral.

Let $f(x)= \lim\limits_{n \to \infty}\dfrac{\cos x}{1+(\arctan x)^n} $ then, Evaluate $\int_0^\infty f(x)\,dx.$ ATTEMPT: Here $$I=\int _o^\infty \lim\limits_{n \to \infty} \frac{\cos x}{1 + ...
0
votes
1answer
16 views

Equivalent definitions of an orthonormal function

I want to prove that the following two definitions for an orthonormal function $\phi$, in terms of $kT$ time shifts, are equivalent. So let $T$ the symbol period and $k$ an integer. Definition 1 ...
0
votes
1answer
37 views

Integrating $\prod_{i=k+1}^{kN} \left(\int_{-\infty}^{\infty}dp_i\right)\times$ with conditions on $p_i$

I am trying to integrate this expression which came up in a derivation of the momentum distribution function for an ideal gas. $\Theta(x)$ is the Heaviside step function which is $1$ when $x$ is ...
16
votes
1answer
194 views

Is it possible to find indefinite integral of $\int \frac{1} {{\sin(x)+\sec^2(x)}}\mathrm{d}x$?

I am in standard $XI$ (i.e.11) and newbie in learning topic of integration. My friend asked me to find indefinite integral of the example shown below $$y=\int \frac{1} {{\sin(x)+\sec^2(x)}} \, ...
4
votes
4answers
243 views

Derivative of a definite integral issue

$g:\mathbb{(0,1]}\to \mathbb{R}$ We have the function $$g\left(x\right)=\int _x^1\left(\frac{\sin\left(t\right)}{t}dt\right)\:$$ Show that the function is strictly decreasing. So I thought that I'd ...
2
votes
0answers
51 views

What are the restrictions on using substitution in integration?

* One photo is equal 1000 words. * Integration done by substitution $u=\tan x$. Integration done by substitution $u=\tan {x\over 2}$. The source function is a continuous positive function ...
3
votes
2answers
134 views

Turning infinite sum into integral

Could someone explain how one can replace infinite sum with integral? Examples (or you may use your own, it doesn't matter as I want to understand principles): $$\frac{1}{n} \sum_{i=1}^{n} \sin ...
-6
votes
4answers
94 views

Easy integral with cube root [closed]

Can you help me? evaluate the following integral: $$\int_0^\frac {\pi} {2} \frac {\sqrt[3]{(sin^2x)}} {\sqrt[3]{(sin^2x)}+{\sqrt[3]{(cos^2x)}}}\,\mathrm{d}x$$ Give me show step by step solutions ...
0
votes
7answers
150 views

Evaluating the indefinite integral $\int \frac{2x+3} {{x^2-2x-3}}\mathrm{d}x$ [closed]

Evaluating an indefinite integral it is simple. Hello it is simple but I can not. Can you help me? evaluate the following integral: $$\int \frac{2x+3} {{x^2-2x-3}}\mathrm{d}x$$ Give me show step by ...
-2
votes
6answers
103 views

Evaluating an indefinite integral with a square root in the denominator

Hello it is simple but I can not evaluate the following integral: $$\int \frac{1} {\sqrt{x^2-2x-8}}\,\mathrm{d}x$$ Give me a clue or show step by step solutions please. Thank you very much.
1
vote
1answer
59 views

What does $d \| \vec{x} \|$ represent?

I am doing some review problems for my upcoming exam and something has come up that I don't understand. $$\int_{\gamma} f(x,y) \space d \|\vec{x} \|$$ $\gamma(t)=(\cos(t),\sin(t))$ ...
2
votes
0answers
86 views

Computation of integral on parametrized curve

if $U$ and $V$ are two open subset of $\mathbb{R}^{n}$, $\varphi:U\rightarrow V$ a $C^{1}$ diffeomorphism, then we have the change of variable formula for the Lebesgue integral: ...
0
votes
0answers
69 views

Continuous and Discrete summation

I have a continuous curve $f(x)$ say constant in interval $0$ to $P$ and area under the curve is unity. Alternately, $f(x) = 1/P$ for $ 0\le x \le P$. So I can calculate $G = \int_{0}^P ...
1
vote
0answers
68 views

Explanation of a Diagram in Wikipedia

In the following diagram (from Wikipedia), there are levels on $y$-axis by red lines. I didn't understand how the rectangles below these red line are drawn? Can one help me? I am thinking the area of ...
1
vote
2answers
61 views

Evaluating simple integrations according to Lebesgue

One difference between Riemann's and Lebesgue's approach of integrations is that Riemann partitions domain of integration, whereas Lebesgue partitions the range of the function. I will consider two ...
0
votes
1answer
44 views

What is the closed form of the given integral?

Does a closed form of integral for $$\int_a^b e^{-c \left( x+\frac{a}{x}\right)} dx$$ exist? How can I approximate such integral? Here, $a=0^+$ (greater than $0$) and c is a positive constant.
-1
votes
3answers
103 views

Integrate $\ln x \cos(\ln x) \,dx$ [closed]

$$\int \frac{\ln x\cdot \cos(\ln x)} {x}\,dx$$ How to calculate this integrate thank you very much
1
vote
2answers
64 views

integral question help me please [closed]

$$\int_0^1 \frac{f(x)} {f(x)+f(1-x)}dx$$ Thank you very much