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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-1
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3answers
390 views

Integral of: $\frac{ 1+\sqrt x}{\sqrt{x-x^2}}$ [closed]

$$\int\frac{1+\sqrt x}{\sqrt{x-x^2}}dx$$ Hi everyone! really desperate about this one. will surely appreciate your help. Thanks:)
0
votes
2answers
72 views

Delta function with variable argument

I've got a double integral involving a delta function, but the complication is that the argument of the delta function depends on both variables that are being integrated over. It's something like ...
4
votes
3answers
159 views

Express $\int_0^1\frac{dt}{t^{1/3}(1-t)^{2/3}(1+t)}$ as a closed path integral enclosing the interval $(0,1)$

From an old complex analysis qualifier: Define $$I=\int_0^1\frac{dt}{t^{1/3}(1-t)^{2/3}(1+t)}.$$ Express $I$ as a closed path integral enclosing the interval $(0,1)$. Evaluate $I$. ...
0
votes
1answer
44 views

Integrating over sums.

I want to make an integral. I know that Integral and Sum can be exchanged. But if I have the following case? $$ \int\left(\sum_{i_1}\sum_{i_2}e^{ix}\right)\sum_{j_1}\sum_{j_2}e^{jx}\,\text dx $$ ...
0
votes
1answer
116 views

Limit $\lim_{n\rightarrow >\infty}\int_I f(x)\cos nx\ \mathrm{d}m(x)=0$

On $\mathbb R$, let $I$ be a finite interval. If $f$ is integrable on $I$, prove that $$\lim_{n\rightarrow >\infty}\int_I f(x)\cos nx\ \mathrm{d}m(x)=0$$ I tried the substitution $nx=u$ but ...
2
votes
2answers
109 views

Integral Question

$$ \int _{\large{ -\frac { \pi }{ 2 } } }^{\large{ \frac { \pi }{ 2 } } }{ \frac { \mathrm{d}x }{ \sin x-2\cos x+3 } } $$ Wolframalpha suggests to evaluate indefinite integral by substituting $u ...
0
votes
3answers
79 views

logarithmic derivative of $x^{e^{(x^2+\cos x)}}$

I'm having a hard time taking the derivative of $$f(x) = x^{e^{(x^2+cosx)}}.$$ I'm aware that I have to take the logarithm of both sides. $$\ln(y) = \ln({x^{e^{(x^2+\cos x)}}}) = \ln(x)\cdot ...
0
votes
2answers
31 views

Equivalent condition to integrability of a function

Let $f$ be a nonnegative Lebesgue measurable function on $[0,1]$. Prove that $f$ is >integrable on $[0,1]$ if and only if $$\sum_{k=0}^{\infty}m(\{x\in [0,1]; f(x)\geq k\}) < \infty$$ where $m$ ...
1
vote
0answers
39 views

How to properly integrate: $\int_0^1 f_i'(y) f_0'(y) dy$

can someone tell me how to integrate: $$\int_0^2 f_l'(y) f_0'(y) dy.$$ I know: $f_l (2)=f_l(0)=0$ I have tried: $ \int_0^2 f_l' f_0'dy=\int_0^2 (f_l f_0' )' dy- \int_0^2 f_l f_0'' dy$ or:$ ...
0
votes
1answer
37 views

Integral Proofs using Darboux or Riemann sums

I have been asked how to prove the following statement: $f$ is integrable $\Longleftrightarrow$ for every $\epsilon > 0$, and every $\mu > 0$ there exists $\delta >0$ such that for every ...
2
votes
3answers
57 views

Integral Of $\oint_{|z-1|=2} \frac{\cos \pi z}{(z^2-z)\cdot (z+2i)}dz$

I want to integrate the following : $$\oint_{|z-1|=2} \frac{\cos \pi z}{(z^2-z)\cdot (z+2i)}dz$$ What I did: $$\oint_{|z-1|=2} \frac{\cos \pi z}{z+2i}\cdot \left [ \frac{1}{z(z-1)}\right]dz$$ ...
2
votes
3answers
111 views

Integration of $\int \frac{dx}{a+f^2(x)}$

I want to solve a integral of the form: $$ \int \frac{dx}{a+f^2(x)} $$ in my particular case I got $$ \int \frac{dx}{5+\cos^2(x)} $$ in my case I followed this process: $$ \int ...
2
votes
3answers
132 views

Finding the improper integral $\int^\infty_0\frac{1}{x^3+1}\,dx$ [duplicate]

$$\int^\infty_0\frac{1}{x^3+1}\,dx$$ The answer is $\frac{2\pi}{3\sqrt{3}}$. How can I evaluate this integral?
0
votes
0answers
45 views

What is the analytical solution to a Volterra integral equation?

I need to solve a following equation: \begin{equation} r_{k+1} = -\sum\limits_{l=0}^{k-1} r_l \cdot (k-l) \cdot \left(\frac{\omega}{t_c - l}\right)^{2 \beta} + \delta_{k,0} \end{equation} subject to ...
1
vote
1answer
99 views

Why is this valid: $2 \int^{\sqrt t}_0 \frac 1 {\sqrt {2\pi}} e^{-u^2/2} du = \frac 1 {\sqrt {2 \pi}} \int^t_0 e^{-s/2}s^{-1/2}ds$

Could someone explain why the following change of variable is valid?: $$2 \int^{\sqrt t}_0 \frac 1 {\sqrt {2\pi}} e^{-u^2/2} du = \frac 1 {\sqrt {2 \pi}} \int^t_0 e^{-s/2}s^{-1/2}ds$$ Using the ...
1
vote
1answer
119 views

integration by trig substitution

I solved the following integral by using trig substitution $(u = 3 sec \theta)$ $$ \int \frac{dx}{x^{2}+10x+16}=\int \frac{du}{u^2-9} $$ and got the same result as in the textbook which looked for ...
2
votes
3answers
67 views

An equivalent of : $f(x)=\int_0^{+\infty}\frac{e^{-xt}}{(1+t^3)^{1/3}} dt$

$\forall\ x\ \in\ \left]0,+\infty\right[\ $ we put: $$ {\rm f}\left(x\right) = \int_{0}^{\infty}{{\rm e}^{-xt} \over \left(1 + t^{3}\right)^{1/3}}\,{\rm d}t $$ The question is the question is to find ...
8
votes
1answer
98 views

How to estimate the limit $\lim_{x\to+\infty}\frac{\int_0^x|\sin(s)|ds}{x}?$

I have come across the problem of estimation of the limit $$\lim_{x\to+\infty}\frac{\int_0^x|\sin(s)|ds}{x}.$$ Since it is easy to check that the method of l'Hopital's Rule is incapable of it. I have ...
1
vote
2answers
66 views

Derivative of an integral with respect to a shifting region

Let $f:\mathbb{R}^3 \to \mathbb{R}$ be a smooth function rapidly decreasing to zero as $|(x,y,z)| \to \infty$, and let $D(t)$ denote $$D(t)=\left\{(x,y,z)\in \mathbb{R}^3 \mid ax+by+cz \leq ...
3
votes
4answers
115 views

slightly tricky integral

was asked to evaluate $\displaystyle\int_0^\infty \dfrac{\log(x)}{1+x^2} dx = I$ firstly, I got the solution using the substitution $ t = \dfrac{1}{x} $ and then getting $\displaystyle\int_0^\infty ...
2
votes
0answers
61 views

Show that $\left| \oint_{\partial D}fdx + gdy \right|^2 \leq (\text{Area}(D))\int_D \left( |\nabla f|^2 + |\nabla g|^2 \right) dx dy.$

Let $\vec{F}=(f,g):\mathbb{R}^2\to \mathbb{R}^2$ be a smooth vector field such that $|\vec{F}(x,y)|\to 0$ rapidly as $|(x,y)|\to \infty$, and let $D$ denote a compact domain in $\mathbb{R}^2$ ...
1
vote
3answers
94 views

How find this integral

$\int_{- \cos x}^{\sin x} \frac{1}{\sqrt{ 1-t^{2}}} dt$ my solution is put $t=sin {\theta}$ then $$\int_{- \cos x}^{\sin x} \frac{1}{\sqrt{ 1-t^{2}}} dt = \int_{- \cos x}^{\sin x} ...
3
votes
2answers
92 views

Finding integration by substitution.

$$ \int_{-\sqrt{\vphantom{\large A}2\,}\,}^{\sqrt{\vphantom{\large A}2\,}\,} x\sqrt{4 - x^{4}}\,{\rm d}x $$ I find the answer $0$ by substitution $x^{2} = u$, but it didn't make sense.
1
vote
3answers
97 views

Let $g$ be a function from $(0, 1)$ to $\mathbb{R}$ and $\int_0^1 g^2(x)dx$ finite. Does this imply that $\int_0^1 g(x)dx$ is finite?

Let $g$ be a function from $(0, 1)$ to $\mathbb{R}$ and $\int_0^1 g^2(x)dx$ finite. Does this imply that $\int_0^1 g(x)dx$ is finite? So I know that $| \int_0^1 g(x)dx | \leq \int_0^1 |g(x)|dx = ...
3
votes
4answers
174 views

Simple integral with a logarithmic function: $\int\frac{\ln x}{\sqrt{1-x}}\,\mathrm dx$

Consider this indefinite integral and solve it with the simplest way. $$\int\frac{\ln x}{\sqrt{1-x}}\,\mathrm dx$$ How can we solve it?
1
vote
1answer
133 views

Build a function $f:[0,1] \rightarrow R$ that is unbounded and Cauchy-Integrable.

Def.: A partition of an interval $[a,b]$ is a set of points $ P=\{x_{0}, x_{1}, ..., x_{n}\}$, such that, $x_{0}=a<x_{1}<...<x_{n}=b.$ Def.: $|P|=max\{x_{i}-x_{i-1} : 1\leq i \leq ...
2
votes
3answers
91 views

Prove that $\int_0^\infty e^{-a^2 s^2} \cos(2 b s) \,\mathrm ds=\frac{\sqrt{\pi}}{2a}e^{-b^2/a^2} $

Prove that $$I(a,b)=\int_0^\infty e^{-a^2 s^2} \cos(2 b s) \,\mathrm ds=\frac{\sqrt{\pi}}{2a}e^{-b^2/a^2}\quad a>0 $$ I can prove it by differential-equations technique(taking the derivative with ...
2
votes
2answers
40 views

How do you integrate this complex function

The integration is a closed integral/circulation (I couldn't find out the mathjax for closed integral sign !) $$ \int_c \frac{-3z+4}{z^2+4z+5}\, dz, \space \space \space \space \space \text {where ...
2
votes
1answer
62 views

Approximating a Gaussian Integral: Can you do better?

I have attempted to approximate this Gaussian: $$ I =\int_{0}^{\lambda}dx\left(r+x\right)\exp\left(-\rho\left(ax^{1/2}+bx^{3/2}\right)\right)\ $$ using $$ I ...
2
votes
2answers
86 views

Find the value of a hard integral [closed]

Find the value of: $$I=\int_0^1\frac{\sqrt{e^x}}{\sqrt{e^x+e^{-x}}} \, dx$$
1
vote
1answer
37 views

Conversion between two types of line integrals

This is probably a silly (and maybe duplicate) question, but I haven't been able to find a standard answer so far. Let $f$ be a real-valued function on $\mathbb{R}^n$, and let $F$ be a function from ...
4
votes
2answers
176 views

Transforming a Riemann Sum to an integral.

Question. Consider the limit $$\begin{align} ...
1
vote
2answers
41 views

Finding in terms of via integration

If $\int_0^{\pi/2}\sin^ {98}x\,\mathrm dx=A$, find $\int_0^{\pi/2}\sin^ {100}x\,\mathrm dx$ in terms of $A$. How can I solve this problem ?
0
votes
1answer
168 views

Evaluate the integral. $\int _\sqrt{2} ^2 x\ln(x^4-1) \, \mathrm dx.$

Evaluate the integral $$\int _\sqrt{2} ^2 x\ln(x^4-1) \, \mathrm dx.$$ As a hint it says that: You can use the fact that $$\int \ln x\,\mathrm dx=x\ln x-x+\mathrm C.$$ at ...
0
votes
1answer
31 views

Dirac delta function of a function - can I make this transformation?

Assuming that $\hat{U}\colon \mathbb{R}^n\to\mathbb{R}$ is continuous and differentiable, is the following step: $\int_{\mathbb{R}^n} \boldsymbol\delta[U-\hat{U}(\mathbf{x})] ...
2
votes
1answer
57 views

Can I switch the order of integration and “Real(z)” operation?

Let $f(z,\eta)$ be an entire function. I need to calculate (numerically) the integral: $$\int\limits _{0}^{\pi}\mbox{Re}\left(\int\limits _{0}^{\pi}f\left(z,\eta\right)d\eta\right)dz$$ Can I switch ...
3
votes
1answer
149 views

Integral Of $\sqrt{9-y^2}dy$

I would like to integrate the following: $$\int\sqrt{9-y^2}dy$$ What I did: $y=3\sin t$ $dy=3\cos tdt$ $$\int\sqrt{9-9\sin ^2 t}\cdot 3\cos t dt=\int\sqrt{9}\cdot\sqrt{1-\sin^2 t}\cdot 3\cos ...
0
votes
1answer
75 views

Scalar surface integral help.

$\iint xy \,\mathrm dS$ where $S$ is the surface of the tetrahedron with sides $z=0$, $y=0$, $x + z = 1$ and $x=y$. The answer is given as: $$\dfrac{3\sqrt{2}+5}{24}$$ ∫∫ xy dS = ∫∫ xy √(1 + ...
2
votes
1answer
59 views

Computing complex integrals

Can anyone help me out with this question: Let $f(z)=\frac{1}{1+z^2}$. Compute $\int_{\gamma_{r}} f(z)dz$ if $r>0$ and $\gamma_{r}$ is the border of {$z \in C: |z| \leq r, Imz\geq 0$}. We walk ...
1
vote
2answers
41 views

Lemma 2.5.5 Boas, Entire functions

I'm reading Boas, Entire functions, but I don't understand lemma 2.5.5, which states that $\sum_{1}^{+\infty}\frac{1}{r_{n}^{\alpha}}$ and the integral $\int_{0}^{+\infty}t^{-\alpha -1}n(t)dt$ ...
3
votes
1answer
1k views

Is this a solution to the indefinite integral of $e^{-x^2}$?

$\int e^{-x^2} \, \mathbb{d} x$, the Gaussian integral, is notorious throughout physics and statistics. Its definite integral defined over $\mathbb{R}$ is $\sqrt{\pi}$. However, the current indefinite ...
-1
votes
1answer
115 views

Integrate $\sqrt{1+(\sin x)^2}$ [closed]

integrate $\sqrt{1+(\sin x)^2}$ how can I solve this in a simple way
18
votes
2answers
684 views

A problem for the New Year

What better to start the year than a dazzling integral? $$\int_{0}^{\infty}\left[1+\left(\frac{2013}{x+2013}+\cdots +\frac{2}{x+2}+\frac{1}{x+1}-x\right)^{2014}\,\right]^{-1}\,dx$$ Happy New Year to ...
0
votes
0answers
204 views

How to bring f(x) from denominator to numerator?

Is it possible to rewrite the following ratio in a way that $f(x)$, its powers or derivatives appears only as numerator. $$\frac{1}{\int_{0}^{c}(c-x)^{2}f(x)dx}$$ $c>0$ is a constant. $f(x)$ is ...
3
votes
4answers
148 views

Riemann-Integral in terms of trapeziums

Why isn't the Riemann integral evaluate in terms of trapeziums ? It would be a better method compared to rectangles.
1
vote
0answers
88 views

Adding truncated normals: calculating convolutions

Problem: Suppose that $X$, $Y$, and $Z$ are independent standard normal random variables. What is the probability of: \begin{equation} P\{ X+Y+Z+\Delta>0 \, | \, Z+\Delta>0, \, ...
0
votes
1answer
37 views

Integration using the divergence theorem.

Let $R=\left \{ \left ( x,y,z \right )\mid x^2+y^2+2z\leq 2, x\geq 0, y\geq 0,z\geq 0 \right \}$ and $\textbf{G}\left ( x,y,z \right )=xyz\left ( 1+\sin ((x^2+y^2+2z)\pi) \right )\left ( x,y,z \right ...
1
vote
1answer
47 views

In the change-of-variables theorem, must $ϕ$ be globally injective?

In the above theorem, doesn't $\phi$ need to be injective too? The inverse function theorem merely implies that $\phi$ is locally injective -- is this sufficient? I ask because Marsden, in his ...
1
vote
1answer
76 views

Integration problem on multiple variables

I would like to ask help on this integration problem: $$\int_0^1\int_0^1\sqrt{x^2+y^2}\;\left[1-\alpha(1-2x)(1-2y)\right]dxdy$$ I was wondering if polar substitution is possible. I have done the ...
0
votes
1answer
53 views

In the change-of-variables theorem, must $\phi (A)$ be open?

In the above theorem, is $\phi (A)$ necessarily an open set in $\mathbb R^n$ ? The author (subsequently) suggests so, but I can't see why. p.s. In the above, "a set has volume" is equivalent to ...