All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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3
votes
5answers
45 views

Find $\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$

How can we find the integral: $$\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$$ I tried to find and got it to be $\cfrac{\pi}{\sqrt2}$. Am I correct? Please help me with an ...
1
vote
3answers
25 views

Solve for $\int \sqrt{x}(\sqrt{x}-2x)^2 dx $

$\int \sqrt{x}(\sqrt{x}-2x)^2 dx $ so I solved this using U-substitution where $u= \sqrt{x}$ so my $du2\sqrt{x}=dx$ then it will be $2 \int u^2(u-2u^2)^2$ and just expand then distribute the $u^2$ so ...
1
vote
1answer
26 views

Application of the mean value theorem for Integrals

Suppose that $f(x)$ is a differentiable function in $[a,b]$, $f^{'}(x)$ is a monotone decreasing function in $(a,b)$, and $f^{'}(b)>0$. So how to prove that $$ \big \vert \int_a^b \cos ...
0
votes
1answer
28 views

Evluating triple integrals via Spherical coordinates

Use Spherical coordinates to evaluate the triple integral $$\iiint_{\mathrm{x^2+y^2+z^2<z}}\sqrt{x^{2}+y^{2}+z^{2}}\, dV,$$ What I tried Converting $x^2+y^2+z^2<z$ to Spherical coordinates ...
4
votes
1answer
345 views

Differentiation under the Integral Sign

Let $X$ be an open subset of $\mathbb{R}$, and $Y$ be a measure space. Suppose that a function $f:X\times Y\rightarrow \mathbb{R}$ satisfies the following conditions: 1.$f(x,y)$ is a measurable ...
1
vote
0answers
7 views

Question regarding double integrals

Regarding the Buffon's needle case for long needles of length $ l>t, $ (the distance between the parallel lines on the floor), we need to solve the integral $$ \int_{\theta=0}^{\frac{\pi}{2}} ...
5
votes
0answers
107 views

Is there closed form for $\int_0^{\pi/4}\exp(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a})\ dx$?

Is there closed form for $$I(a)=\int_0^{\pi/4}\exp\left(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a}\right)dx $$where is $a\in (-1,3)$ I've tried with $\tan x=u$ and I got the result of sum in term of ...
-4
votes
1answer
34 views

How to calculate this kind of integral?

General form: $$ \int \sqrt{\alpha x^3 + \beta x^2 + \gamma x + \delta} \, dx $$ Example: $$ \int \sqrt{\frac{2}{3} x^3 + x^2 + 4} \, dx $$ Please describe more details.(I'm a freshman.) I will ...
9
votes
4answers
141 views

Integral: $\int_0^{\pi/12} \ln(\tan x)\,dx$

I am trying to evaluate: $$\int_0^{\pi/12} \ln(\tan x)\,dx$$ I think the integral is quite simple but I am having a hard time evaluating it. I started with the result: $$\int_0^{\pi/4} \ln(\tan ...
3
votes
4answers
144 views

Inverse Trigonometric Integrals

How to calculate the value of the integrals $$\int_0^1\left(\frac{\arctan x}{x}\right)^2\,dx,$$ $$\int_0^1\left(\frac{\arctan x}{x}\right)^3\,dx $$ and $$\int_0^1\frac{\arctan^2 x\ln x}{x}\,dx?$$
2
votes
0answers
44 views

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
7
votes
0answers
43 views

Real analytic methods for the following integral

A few days back, the following integral was posted $$\int_0^1 x^x(1-x)^{1-x}\sin(\pi x)\,dx=\frac{\pi e}{24}$$ The integral was answered using complex analysis tools but I am interested in other ...
0
votes
2answers
22 views

feedback on my solution (improper integral)

i have done this improper integral but i am not sure if i have followed the correct procedure or my answer is correct. Please help!
4
votes
3answers
216 views

Show that $\int_0^{2\pi}\frac{R^2-r^2}{R^2 - 2Rr\cos (\varphi-\vartheta) + r^2}d\vartheta$ is independent of $R>r>0$, using only real numbers.

The poisson kernel is sometimes written as $$ \frac{1}{2\pi}\int_0^{2\pi} \frac{R^2-r^2}{R^2 - 2Rr\cos(\varphi-\vartheta) + r^2} \mathrm{d}\vartheta = 1 \ , \ \ R>r>0 $$ Where $\varphi$ is some ...
0
votes
1answer
10 views

Joint CDF from conditional cdf

I would like to derive an expression of the following joint CDF $P[X \leq x,Y \leq y]$ based on the conditional CDF $P[X \leq x | Y=y]$ and the pdf $P[Y=y]$ that are considered to be known. I get a ...
4
votes
3answers
120 views

Integrate $\,\displaystyle\int_0^{\infty } \frac{\cos x}{x} dx$

Although I have known that $\displaystyle\int_0^\infty {{\sin x} \over x} \, dx = {\pi \over 2}$, I have no idea how to work out $\displaystyle\int_0^{ + \infty } {{\cos x} \over x} \, dx$. How can ...
2
votes
1answer
25 views

feedback on my solution (integration)

I need help in this problem. I managed to find the answer for this problem by using mathmatica but cannot do the working for it. i have done most of it but i am stuck on the last part.
-2
votes
0answers
32 views

how to calculate the following definate integral [on hold]

∫f(a,b,x)dx= Antiderivative or integral could not be found.
6
votes
0answers
61 views
+50

Closed form of $\int_0^1 \operatorname{Li}_3^3(x)\,dx$ and $\int_0^1 \operatorname{Li}_3^4(x)\,dx$

We know a closed-form of the first two powers of the integral of trilogarithm function between $0$ and $1$. From the result here we know that $$I_1=\int_0^1 \operatorname{Li}_3(x)\,dx = ...
8
votes
1answer
77 views

Closed form of $\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx$

Inspired by this question, is there a closed-form of $$\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx\,?$$ Here $n \in \mathbb{N_+}$. In the answers to the question above we could find proofs of ...
1
vote
2answers
45 views

Integration of $F(\sum_k x_k)$ over positive orthant

Problem Suppose we some function $F\left(\sum\limits_{k=1}^n x_k\right)$ over the positive orthant $[0,\infty)^n$. Show that this this is proportional to the integral $\int\limits_0^\infty ...
0
votes
0answers
31 views

A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Proof:$\forall a,b;c,d\in\mathbb{R},a<b,c<d.$ $f (x+y) $ is ...
-8
votes
0answers
34 views

Please solve this basic question [on hold]

integral of ${sec}^ \theta\, dx$ "i need the answer immediately"
2
votes
1answer
49 views

Really tricky integration-----double U and trig substitution

The definite integral $h(x) = \sin x/(1 + x^2)$ on the closed interval $[-1,1]$ $\tan^2(x) + 1 = \sec^2(x)$, $x = \tan(@)$ $$x = \tan(@)$$ $$dx = \sec^2(@) d@$$ now I have to find sin9x0 in terms ...
1
vote
2answers
37 views

How do I approach this double integral?

Let $R$ be the region inside $$x^2+y^2 = 1$$ but outside $$x^2+y^2 = 2y$$ with $x \ge 0 $ and $y \ge 0$ Let $$u = x^2 + y^2$$ and $$v = x^2+ y^2 - 2y$$ Compute $ \iint_R xe^y dxdy$ using this change ...
2
votes
1answer
116 views

The equivalence of an integral and a sum of integrals

Let $p(x)$ be a polynomial. Assume that $ \displaystyle \int_{a}^{b} p(x) \cot \left(\frac{ax}{2} \right) \ dx $ converges. Then $$ \int_{a}^{b} p(x) \cot \left(\frac{ax}{2} \right) \ dx = 2 ...
0
votes
0answers
12 views

integration featuring the unit step function

Compute the following integrals I don't know how to use MathJaX so here's a link to the image of the integrals where u(t) is the unit step function and σ is some variable of integration
0
votes
1answer
30 views

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$ I integrated it once to get, $2x + \sin x + C$, $C$ being a constant. Then I integrated it a second time to get $x^2 - \cos x ...
0
votes
0answers
15 views

Integration by parts partial derivatives

Given $$\int_x \int_t \Big( \frac{\partial}{\partial t}u(x,t) + \frac{\partial}{\partial x}f(u(x,t)) \Big) \phi(x,t)~~ dt dx = 0$$ How can I apply integration by parts in order to have the ...
1
vote
1answer
38 views

Calculate $\int_0^1f(x)dx$

Calculate $\int_0^1f(x)dx$,where $$\ f(x) = \left\{ \begin{array}{l l} 0 & \quad \text{if $x=0$ }\\ n & \quad \text{if $x\in(\frac{1}{n+1},\frac{1}{n}]$} \end{array} \right.$$ ...
0
votes
0answers
48 views

Can this be expressed by a contour integral?

Let $f(z)$ be a real entire function of the form $f(z) = a_1 z + a_2 z^2 + ...$ such that $0 < a_{n+1} < a_n$. Consider $g(x) = f^{-1}(f(x)-f(x-1))$ where $x$ is a positive real and $f^{-1}$ ...
3
votes
3answers
56 views

How to compute $ \int e^{-st} \sin(2t) dt $

Wolfram Alpha shows me the result of $ \int e^{-st} \sin(2t) dt $ . However it doesn't let me see the step to step solution. Then I tried to do this by hand as the solution didn't look "too ...
0
votes
1answer
24 views

Integrate $\int \csc 2Q\,\mathrm{d}Q$

I need to use $\cot Q+\tan Q=2\csc 2Q$ to integrate $$\int \csc 2Q\,\mathrm{d}Q.$$ the integral becomes $$\frac12\int\left(\frac{\cos Q}{\sin Q} + \frac{\sin Q}{\cos Q}\right)\,\mathrm{d}Q$$ ...
2
votes
3answers
79 views

If $f'(x)=f(x)+\int_{0}^{1}f(x)\,dx$ and $f(0) = 1,\,$ then what is the value of $\, \int_0^1 f(x)\,dx=$?

If $\displaystyle f'(x)=f(x)+\int_{0}^{1}f(x)\,dx\,$ and $\,f(0) = 1.$ Then what is value of $\displaystyle \int f(x)\,dx\,?$ $\bf{My\; Try.}$ Let $\displaystyle \int_{0}^{1}f(x)\,dx = A\;,$ Then ...
0
votes
1answer
24 views

Is there a clever way to determine negative area of an integral?

Given some continuous, integratable function f(x) that has only positive area over a range from x1 to x2...is there a way to determine the negative area of the integral of f(x) - c (from x1 to x2), ...
1
vote
1answer
321 views

Approximating integrals with step functions

For $f \colon [1,2] \to \mathbb{R}$ , $f(x) = 1/x$, Choose a sequence of step functions $\phi_n$ approximating $f$ with partition $P_n := [r/n : n < r < 2n]$ to show that $ 1/(n+1) + \cdots + ...
0
votes
2answers
34 views

Spectral Measures: Support vs. Norm

Given a complex Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ and its associated normal operator: $$T:=\int_\mathbb{C}zdE(z)$$ ...
13
votes
5answers
195 views

How to proof the following function is always constant which satisfies $f\left( x \right) + a\int_{x - 1}^x {f\left( t \right)\,dt} $?

Suppose that $f(x)$ is a bounded continuous function on $\mathbb{R}$,and that there exists a positive number $a$ such that $$f\left( x \right) + a\int_{x - 1}^x {f\left( t \right)\,dt} $$ is constant. ...
0
votes
1answer
42 views

Spectral Measures: Riemann-Lebesgue

Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. Consider a Borel spectral measure $E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$. Denote its associated ...
0
votes
3answers
39 views

Spectral Measures: Support vs. Spectrum

Given a complex Hilbert space $\mathcal{H}$. Consider a Borel spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ and its associated normal operator: ...
0
votes
0answers
11 views

ODE - Laplace transform

I have an ODE $\psi^{'}(s)_{3 \times 3}=(A+Bs)_{3 \times 3}\psi(s)_{3 \times 3} \tag1$ where A,B are constant skew symmetric matrices with zero determinant. $\psi(s)$ is a rotation matrix. It implies ...
7
votes
2answers
146 views

Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$

At first sight it looks like the integral below $$\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$$ can be evaluated by using some geometric series. What else can we do? Is there a fast easy ...
2
votes
1answer
41 views

Compute a multiple integral$\iint_{[0,1]^2} (xy)^{xy} dxdy$

$$\text{Compute} :\iint_{[0,1]^2} (xy)^{xy} dxdy$$ I am thinking about changing the variable, $x=u,y={v \over u}$.But it doesn't work. I just found that the answer is$\int_0^1 t^t dt$.Maybe my idea ...
3
votes
3answers
202 views

Finding the Integral of a function

$\int \dfrac{1}{9+x^2}dx$ The answer is $\arctan(x/3)/3 + C$ , but I don't understand the process of how the answer was found. I tried using u-substitution, but I came up with 2xdx/x^2+9.
2
votes
5answers
630 views

How can I show that these integrals are zero

How can I show that these integrals equal $0$ when $n$ and $m$ are both integers and $n \neq m$? $$\int_{-\pi}^{\pi}\sin(mx)\sin(nx)dx = \int_{-\pi}^{\pi}\cos(mx)\cos(nx)dx = 0$$ I'm able to show that ...
2
votes
0answers
21 views

If $f$ is increasing, then for all $n\in\mathbb{N}$ there exists $P_n$ : $U(f,P)-L(f,P) \leq (b-a)/n$

I've already proven that, if $f:[a,b] \to \mathbb{R}$ is continuous and increasing, with $a,b\in \mathbb{R}$, then $$U(f,P) - L(f,P) = \sum_{i=1}^{n}\left[ f(x_i) - f(x_{i-1})\right](x_i - x_{i-1})$$ ...
4
votes
1answer
52 views

Is there an alternative way to solve this integral?

I was given the integral $$\int \frac{2}{e^{-x}+1}dx$$ Here is my method to get the (correct) solution: $$\int \frac{2}{e^{-x}+1}dx$$ $$=2\int \frac{1}{e^{-x}+1}dx$$ $$=2\int ...
8
votes
2answers
513 views

Integral of rational function with higher degree in numerator

How do I integrate this fraction: $$\int\frac{x^3+2x^2+x-7}{x^2+x-2} dx$$ I did try the partial fraction decomposition: $$\frac{x^3+2x^2+x-7}{x^2+x-2} = \frac{x^3+2x^2+x-7}{(x-1)(x+2)}$$ And: ...
0
votes
0answers
55 views

What is the recurrence relation between $a_n, a_{n-1}$ , $a_n = \int_0^1 {x^n}\tan\left( \frac{\pi}{4}x\right) dx$

I would appreciate if somebody could help me with the following problem Q: What is the recurrence relation between $a_n, a_{n-1}$ ? $$ a_n = \int_0^1 {x^n}\tan\left( \frac{\pi}{4}x\right) dx,\ \ ...
8
votes
2answers
191 views
+50

Calculation of $\int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$

Calculate $$ \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2\cos^2 ...