Find the integral   $$\int\int _{[0,1] \times[0,1]} \max \left\{x,y\right\} dx \,dy$$
123391
Evaluate:  $$\int_{[0,1]×[0,1]} \max\{x, y\} dxdy$$     I am totally stuck on it. How can I solve this?
11850
If we define on $C([0,1])$ the operator $$ Tf(x) = \int_{0}^{1} K(t,s) f(s) ds$$ where $K$ is a continous function on two variables. I want to show that:  $1)$ $||T|| = \displaystyle\max_{t} {\int_{0}^{1} |K(t,s)| ds} $  $2)$ When $K(t,s) = \displaystyle\min(t,s)$, to prove that $T$ is compact  Thanks.
106191
\begin{eqnarray*} M&amp;=&amp; \max{\left \{a,b,\frac{a+b}{2},\right}\\ &amp;&amp; a+b,a+b/2\Bigg\} \end{eqnarray*}
17488
In a certain list of exercises, contained the question     $$\int_{0}^{\infty}{\left(x^3\cdot(e^x-1)^{-1} \right)}dx$$   How do you solve?
36811
Suppose $K(x,y)$ is a symmetric kernel. Let $\phi\in L^2(\Omega)$, where $\Omega$ everywhere is a domain in $R^n$.  Can $\int_{\Omega}K(x,y)\,\phi(y)\,dy$ belong to $L^2$?  In other words can an inequality of the type  $$ \left\|\int_{\Omega}K(x,y)\,\phi(y)\,dy\right\|_{2} \leq \|K\|_{L^1(\Omega\times\Omega)} \|\varphi\|_{2} $$ exist?
71139
Find the maximum value of $72\int\limits_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx  $ for $y\in[0,1].$  I tried to differentiate the given function by using DUIS leibnitz rule but the calculations are messy and I tried to solve directly by integrating it but that also is not working.Can someone please help me in solving this question?
57059
I came across this problem :  Let $f\colon [a,b]\rightarrow \mathbb{R}^n$ a continuous vector valued function. Then it is true that:  $$\left\Vert\int \limits_a ^b f(t) dt\right\Vert \leq \int \limits_a^b \left\Vert f(t)\right\Vert dt$$  I consider that the integral of $f$ is given by integrating each component and then I am trying to reduce it to having an integral inequality for $n$ single valued functions but I am stuck.  I would appreciate some help.  Thank you!
18467
Let $f(x)$ be a continuous real-valued function on $[a,b]$ and $M=\max\{|f(x)| \; :\; x \in [a,b]\}$. Is it true that: $$ M= \lim_{n\to\infty}\left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n} ? $$  Thanks!
100942
$$ \int_{0}^{\infty}\frac{\mathrm{d}x}{\left(x^{2} + 1\right)\left[\,{\ln^{2}\left(x\right) + \pi^{2}}\,\right]} $$ Calculate the following integration using the residue theorem.
109939
Let $f$ be a continuous function. I wish to show $$\lim_{n \rightarrow \infty} \int_0^1 n x^n f(x) \; dx = f(1)$$ I can try to split up the integral over intervals $[0, 1-\delta]$, $[1-\delta, 1]$. The integral over $[0, 1-\delta]$ vanishes as $n \rightarrow \infty$, but I am having difficulty estimating the integral over $[1-\delta, 1]$. Thanks!
45881
Show that $\int_{0}^{1}\int_{0}^{1}(xy)^{xy}dxdy = \int_{0}^{1}y^{y}dy$  I have tried the method used in the Gaussian integral, polar coordinates, exp(.) and ln(.) of both sides of the integrand...  Gaussian integral:   I have also looked at the sophomore dream function, but I am not sure how to proceed that way in this case.  Sophomore dream function:    Now I am stuck. Any help would be appreciated.
96419
$$P\left(Z\le z\right)=P\left(\left|X-Y\right|\le z\right)=P\left(-z\le X-Y\le z\right)=P\left(Y-z\le X\le Y+z\right)$$ This means that, because $\space f(x,y)=f(x)f(y) \space$ as they are independent, we get to calculate this integral: $$\int _0^{\infty }\:\int _{y-z}^{y+z}\:\left(\lambda e^{-\lambda x}\right)^2dxdy$$ We get $$\left(e^{\left(-2\cdot \lambda \cdot z\right)}\cdot \left(e^{\left(4\cdot \lambda \cdot z\right)}-1\right)\right)/4$$ Which is not what we want So where's the mistake?
86584
How can I solve integral of types  (1) $\displaystyle \int\dfrac{1}{\sqrt[4]{x^4+1}}dx$  (2) $\displaystyle \int\dfrac{1}{\sqrt[4]{x^4-1}}dx$
26140
Question:  If $f\in C^1[0,1]$, show that  $$\int_0^1 |f(x)|dx\le\max\left\{\int_0^1 |f&#39;(x)|\,dx,\;\bigg|\int_0^1 f(x)\,dx\bigg|\right\}.$$  I have tried to make connection between $|f|$ and $|f&#39;|$ by using  $$(tf(t))&#39;=f(t)+tf&#39;(t),$$  integrate the equation with $t$ on $[0,1]$, it gets  $$|f(1)|\leq \bigg|\int_0^1 f(t)\,dt\bigg|+\int_0^1 |f&#39;(t)|\,dt,$$  but it is not the desired inequality.
84554
What is $$\int\Gamma(x) \, dx$$ where $\Gamma(x)$ is the gamma function?
109003
What is the value of $$\int_0^{\pi} e^{a\cos(t)}\cos(a\sin t)dt?$$
47826
Evaluate the following integral $$\iint_S (x+y+z) \, dS$$  where $S$ is the surface of the cube $[0,1] \times [0,1] \times [0,1]$  Honestly, I don't know what to do. All I know is that you have to evaluate the integral for each of the 6 surfaces of the cube. The normals for them I can do and the equations of the planes, but other than that I couldn't do much.  Can someone break it down for me, please?  PS: No divergence theorem yet for me.