What is the integration of $$\int x^x dx$$ And how can I understand whether an integration is possible or not? Is there any rule to understand whether a function is integrable or not?
13823
I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this:  Let $u=x^x$.  $$\begin{align} \int x^x \, dx&amp;=\int u \, du\\[6pt] &amp;=\frac{u^2}{2}\\[6pt] &amp;=\dfrac{\left(x^x\right)^2}{2}\\[6pt] &amp;=\frac{x^{2x}}{2} \end{align}$$  But it's certain that this isn't the correct way to evaluate that, and the answer must be wrong.
35060
Can I get help to prove or Disprove that -   $\int_{-a} ^a F_1(x)F_2(x)dx = 0 \ for \ a &gt; 0$  when I know that $F_2$  is even function and $F_1$  is odd.  I would like to get the idea how to delete and extract Minus in integrals.  thanks!
3572
The question:      Compute$$ \int \frac {x^2 \, \operatorname{d}\!x} {(x\sin x+\cos x)^2} $$   Tried integration by parts. That didn't work.  How do I proceed?
119970
$\displaystyle\int_0^0\frac 1x\:dx$ This integral is defined ? If it define, what is the value ?  Please,step by step
43973
I am working on a few examples where my author said this is true if $h$ is continuous. Can someone explain why? Just because the integral is $0$ doesn't mean the integrand is zero as well.   $\int_{a}^b h^2=0\implies h=0$?
88681
Prove that if $f:[a,b] \rightarrow \mathbb{R}$ is integrable, then so is $|f|$ and $\left|\int_{a}^{b}f(x)dx\right| \leq \int_{a}^{b}|f(x)|dx$   I already proved that |f| is integrable, but how do I show that $|\int_{a}^{b}f(x)dx| \leq \int_{a}^{b}|f(x)|dx$   ?
29793
Assmue that $f(x)\in C(0,+\infty)$ and for any $x&gt;0$ have $$\int_{x}^{x^2}f(t)dt=\int_{1}^{x}f(t)dt$$  Find this $f$  My try: we have $$2xf(x^2)-f(x)=f(x)\Longrightarrow xf(x^2)=f(x),\forall x&gt;0$$ and then I can't works
77350
$\forall x \in \mathbb R^d$ and $m &gt; d/2$.  I know that $\int_{\mathbb R^d}{(1+|x|^{2})^{-m}dx}$ is integrable, but how can I prove?  Thanks!
64675
$$\int_{0}^{\infty }\sin(x^{2})dx$$  I'm confused with this integral because the square is on the x, not the whole function. How can I integrate it? Thank you.  I have not done complex analysis (only real analysis as I am a high school student) so how can I evaluate it using elementary functions (without complex analysis)?
36902
How to find integral of: $$\int xe^x\sin(x)$$ I know it should be repeated integration by parts, but i do not know how many times i should do this and when to stop.
74257
$$\int \sqrt{\sin x}\ \text dx$$   I asked my teachers and they said that this integration is pretty next level and will be taught in college. Can anyone help?
104960
I am curious about the proof of these two integrations. It would be great if you could point me in the right direction : $ \int{e^{ax}\cos({bx})dx}= \dfrac{e^{ax}}{a^{2}+b^{2}}[a\cos(bx)+b\sin(bx)] + C $ &amp; $ \int{e^{ax}\sin({bx})dx}= \dfrac{e^{ax}}{a^{2}+b^{2}}[a\sin(bx)-b\cos(bx)] + C $
105184
Hi consider the following expression:  $F(x)=\int_a^{x}f(t)(x-t)dt$ and hence determine $F'(x)$.  This is what I have I done: first split into  $F(x)=\int_a^{x}xf(t)dt$ - $\int_a^{x}tf(t)dt$  The second half of the expression is integral of an expression fully interms of t, hence I know by the fundemental theorem of calculus upon differentiating it i can rewrite it as $xf(x)$.  However I have problem getting past the first half $\int_a^{x}xf(t)dt$ it is an integral containing x and t. is there any technique that I can apply in order to evaluate this integral?
74195
Good afternoon, I need to find:  $$\int_0^\infty\frac{x^n}{ax^2+bx+c}\space dx$$  I've no idea about any clue what so ever.
3739
I want to prove:     For an integrable function $f(x)$ and periodic with period $T$, for every $a \in \mathbb{R}$, $$\int_{0}^{T}f(x)\;dx=\int_{a}^{a+T}f(x)\;dx.$$   I tried to change the values and define $y=a+x$ so that $dy=dx$ and the limits of the integrals are as we want, but I'm not sure how to use the fact that $f(x)$ is periodic.  Thanks a lot!
108075
The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval.  (Note, it is not absolutely Riemann Integrable.)  Why is it we restrict our definition of Lebesgue Integrability to absolutely integrable?  Wouldn't it be better to extend our definition to include ALL cases where Riemann Integrability holds, and use the current definition as a corollary for when the improper integral is absolutely integrable?
54134
Integrate $\displaystyle \int \limits_0^\pi{{x\sin x}\over{1+\cos^2x}}dx$. I tried substituting $t=\cos x$, and then integrate with integration by parts. It got all messy... Thanks in advance for any help!