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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4
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3answers
123 views

How to integrate $ \int_0^\infty \sin x \cdot x ^{-1/3} dx$ (using Gamma function)

How can I calculate the following integral: $$\int_0^\infty x ^{-\frac{1}{3}}\sin x \, dx$$ WolframAlpha gives me $$ \frac{\pi}{\Gamma\Big(\frac{1}{3}\Big)}$$ How does WolframAlpha get this? I ...
4
votes
5answers
101 views

Is it true that $\int\frac{1}{x}\,dx=\ln x\implies\ln x = \frac{x^0}{0}$?

I have recently learned that $\int\frac{1}{x}\,dx=\ln x$. However, by the power law of integration, the integral of $1/x$ is equal to $x^0 / 0$ which is undefined. Therefore, is $x^0 / 0 = \ln x$? ...
4
votes
2answers
110 views

How to find the definite integral $\int_0^\infty \frac{x}{\sinh ax}\;dx$

I'm trying to prove that $$I:= \int_0^\infty \frac{x}{\sinh(ax)} dx = \frac{\pi^2}{4a^2}$$ Attempt: $$\sinh (ax) = \frac{1}{2}(e^{ax}-e^{-ax}) = \frac{1}{2}e^{-ax}(e^{2ax}-1)$$ Now I have ...
4
votes
2answers
144 views

Definite Integrals problem

The question is to find the value of : $$\frac{\displaystyle29\int_0^1 (1-x^4)^7\,dx}{\displaystyle4\int_0^1 (1-x^4)^6\,dx}$$ without expanding. According to the book, the answer is 7. I tried taking ...
4
votes
2answers
78 views

Evaluate $\int\frac{\cot x}{\cos^2 x-\cos x+1}\,\,dx$

$$\int\frac{\cot x}{\cos^2 x-\cos x+1}\,\,dx$$ Please guide me by which term it should be substituted to get the result of this integration. I have tried it by using $\cos x =t$, but it went so long ...
4
votes
5answers
183 views

How Prove this integral is diverge $\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$

Show that this following integral is divergent (or diverges, if you prefer) $$\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$$ I know when $x=0,1$ are singularities of the function and I want use this ...
4
votes
2answers
87 views

How to compute $\int_0^{\infty} x^{t-1} e^{-x}\ln(x)\,dx$?

I have hit the following integral (in the process of trying to derive a finite-sample correction for the Maximum Likelihood fitting of the Generalized Extreme Value distribution...): ...
4
votes
2answers
147 views

How to find this integral $I=\int_{-\pi}^{\pi}\frac{x\cdot \sin(x) \cot^{-1}{(2014^x)}}{1+\cos^4(x)}dx$

Question: Find this integral $$I=\int_{-\pi}^{\pi}\frac{x\cdot \sin(x) \cot^{-1}{(2014^x)}}{1+\cos^4(x)}dx$$ let $x\to -x$,so $$I=\int_{-\pi}^{\pi}\dfrac{x\sin(x) ...
4
votes
3answers
88 views

Integrate $\ln(2-x)dx$

I want to learn how to integrate this. If you could show me a step-by-step approach that would be awesome. If you could also point me to some good tutorials on integration that would be icing on the ...
4
votes
1answer
192 views

Can we express the following in a closed form? [duplicate]

I want to evaluate the integral: $$I=\int_{0}^{\pi/2}\ln \left ( \frac{(1+\sin x)^{1+\cos x}}{1+\cos x} \right )\,dx$$ Well, the sub $u=\pi/2-x$ does not give me any result. In fact it makes the ...
4
votes
2answers
117 views

Apostol (6.22.10): Finding $\int \frac{\arcsin x}{x^2} dx$

I'm trying to solve another integral from Apostol (Chapter 6, Section 6.22, Question 10) which says to show the following: $$ \int \frac {\arcsin x}{x^2}dx = \log|{\frac {1-\sqrt{1-x^2}}{x}}| - \frac ...
4
votes
4answers
176 views

How to integrate: $\int_{0}^{\infty}e^{tx}(x^2e^{-x})/2dx$

I'm working on a few moment generating function problems and I came across: $f(x)=(x^2e^{-x})/2$ for $x>0$, and zero otherwise. Find the mean. The mean is the same as the expected value. If we ...
4
votes
3answers
97 views

Explanation of $\int_0^{2\pi}\sin^{100}x\,dx$.

I was browsing this thread when I came across this answer. I can neither make heads nor tail of it. Can someone help me understand it? This I understand: ...
4
votes
1answer
226 views

A hard problem on exponential integration

Suppose $a : [0 , 1] \to \Bbb R$ is an infinitely smooth function. For $\lambda\ge1$, define $$F(\lambda) := \lambda \int_0^1 e^{\lambda t} a(t) \, dt.$$ If $\sup_{\lambda\ge1}|F(\lambda)|\lt\infty$, ...
4
votes
4answers
69 views

Simplifying a u-substitution for $\int \frac{x} { \sqrt {4-3 x^4 } } \, dx$

this is a calculus one problem I cannot figure out. I may be making a simple assumption in my substitutions, please help. (I hope I typed this correctly, this is my first time using the MathJaX ...
4
votes
1answer
118 views

Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $

Need to prove $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $ and $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{y}{1+y^2}e^{-ay} dy =0 $ Can ...
4
votes
2answers
129 views

The antiderivative of $\sin(1/x)$

How to prove that the function $f(x)=\sin\frac{1}{x}$ for $x\neq 0,f(0)=0$ has an antiderivative? This means $F(x)=\int^{x}_{0}\sin(1/t)dt$ has derivative $0$ at $x=0$, but I have no idea how to prove ...
4
votes
4answers
710 views

Calculating $\int_0^\infty \frac {\sin^2x}{x^2}dx$ using the Residue Theorem.

I am trying to compute the following integral using the Residue Theorem but am quite stuck: $$\int_0^\infty \frac{\sin^2x}{x^2}dx$$ I have tried applying Jordan's lemma, having written $\sin(x)$ as ...
4
votes
3answers
58 views

Calculus and minimum values

This is a simple question but I think I don't understand exactly what the question is asking.
4
votes
3answers
364 views

Evaluating $\int_0^1 \int_0^{\sqrt{1-x^2}}e^{-(x^2+y^2)} \, dy \, dx\ $ using polar coordinates

Use polar coordinates to evaluate $\int_0^1 \int_0^{\sqrt{1-x^2}}e^{-(x^2+y^2)} \, dy \, dx\ $ I understand that we need to change $x^2+y^2$ to $r^2$ and then we get $\int_0^1 \int_0^{\sqrt{1-x^2}} ...
4
votes
3answers
92 views

Integrating $\int^{e^3-1}_{0}\frac{dt}{1+t}.$

How can I integrate $$\int^{e^3-1}_{0}\frac{dt}{1+t}.$$ I tried to make $u=1+t$ which means that $du=dt$ but it's not giving me anything useful, but instead made things more complicated. Maybe I did ...
4
votes
7answers
3k views

LIATE / ILATE rule

In another question of mine users proposed the LIATE or ILATE rule for partial integration. However, I have encountered a problem: $$ e^{-x}\cos(x)$$ If I use the rule, I eventually get: $$ ...
4
votes
3answers
328 views

How to prove that $\lim\limits_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0\text { ? }$

Let $f:\left[a,b\right]\to\mathbb{R}$ be a function that is derivative so that $f'$ is continuous then $$ \lim_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0 $$ My attempt: I ...
4
votes
2answers
136 views

Find the value of $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+xy+y^2)}dx\,dy$

Given that $\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}$. Find the value of $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+xy+y^2)}dx\,dy$$ I don't understand how I find this double ...
4
votes
2answers
135 views

How to calculate this improper integral $\int_0^{+\infty} e^{-(ax+\frac{b}{x})^2}\mathrm{d}x$?

How to calculate this improper integral $$ \int_{0}^{\infty}{\rm e}^{-\left(ax\ +\ b/x\right)^2}\,{\rm d}x\ {\large ?} $$
4
votes
2answers
203 views

Need some hint or technique for this integral

What technique we should use for this integral: $$\int_0^1\frac{\ln x\ln (1-x^2)}{1-x^2}\text{d}x$$ Can anyone give a brief way to evaluate this?
4
votes
2answers
347 views

integration inequality [duplicate]

Possible Duplicate: Proving Integral Inequality Suppose $f(x)$ is differentiable on $[0,1]$ , $f(0)=0$ and $1\geq f'(x) >0 $ Prove that $\displaystyle\left(\int_{0}^{1} ...
4
votes
2answers
320 views

Integration of $x^3 \tan^{-1}(x)$ by parts

I'm having problem with this question. How would one integrate $$\int x^3\tan^{-1}x\,dx\text{ ?}$$ After trying too much I got stuck at this point. How would one integrate $$\int ...
4
votes
1answer
176 views

How to integrate $\int\frac{x}{1+x^3}dx$?

How to integrate $\displaystyle \int\frac{x}{1+x^3}dx$? I tried using partial fractions and substitution but it didn't work, thanks.
4
votes
4answers
199 views

Evaluating $\int \frac{l\sin x+m\cos x}{(a\sin x+b\cos x)^2}dx$

How do I integrate this expression: $$\int \frac{l\sin x+m\cos x}{(a\sin x+b\cos x)^2}dx$$.I got this in a book.I do not know how to evaluate integrals of this type.
4
votes
1answer
233 views

Bounding $\int_0^1 f(x) dx$, given $\int_0^1 f'(x)^2 dx \leq 1$ and $f(0) = 0$.

Let $S$ be the set of all differentiable function $f \colon [0,1] \rightarrow \mathbb{R}$ such that $\int_0^1 f'(x)^2 dx \leq 1$ and $f(0) = 0$. Define $J(f) := \int_0^1 f(x) dx$. Show that $J$ is ...
4
votes
2answers
2k views

how to solve double integral of a min function

I came across this integral in a book. I don't understand how the author writes the following expression for it. $$\int_0^T \int_0^T \min(t,s)\, dt\, ds = \int_0^T \left(\int_0^s t\, dt + \int_s^T s ...
4
votes
1answer
1k views

How can I evaluate the integral of $\exp(-x^2-1/x^2)dx$ from 0 to infinity?

Question : integral of $\exp(-x^2-1/x^2)dx$ from $0$ to infinity I think the answer is square root of $ \pi/2 \cdot \exp(-2)$. If I change $-x^2-1/x^2$ to $-(x-1/x)^2-2$ then the above integral ...
4
votes
2answers
515 views

Digamma function integral

Does anyone how to get a finite value to this integral ? $ \int_{0}^{\infty} dx \frac{ \Psi (1/4+ix/2) +\Psi (1/4-ix/2)}{x^{2}+1/4} $ i have tried residue theorem but i got nonsenses :( can anyone ...
4
votes
2answers
9k views

Finding the volume of a sphere with a triple integral and trig sub

How is trigonometric substitution done with a triple integral? For instance, $$ 8 \int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} (1) dz dy dx $$ Here the limits have been chosen to ...
4
votes
2answers
82 views

Find a function $f(x)$ in an integral

(Related question here). Is there a way to calculate the function $f(x)$ in this integral in terms of $x$ without using $a,b,c$: $$\int_{a}^{b} f(x)dx=c$$ Two examples $\rightarrow$ how do find ...
4
votes
2answers
77 views

Can anyone help me with this improper integral?

$$\int_{0}^{\infty} \left(e^{-\frac{1}{x^2}}-e^{-\frac{4}{x^2}}\right) dx$$ I've tried much of the techniques used in the textbook, none have led to anything concrete, or i am not just able to see ...
4
votes
3answers
88 views

Show that: $\int_{0}^{\infty} \frac{\sin{x^{q}}}{x^{q}} dx = \frac{\Gamma{\frac{1}{q}}}{q-1}\cos{\frac{\pi}{2q}} \mbox{, q > 1}$

How do you show that: $$ \int_{0}^{\infty} \frac{\sin{x^{q}}}{x^{q}} dx = \frac{\Gamma{\frac{1}{q}}}{q-1}\cos{\frac{\pi}{2q}} \mbox{, q > 1} $$ Without using Gamma function?
4
votes
2answers
253 views

Integrating a jacobian to find the volume.

I want to solve the following: Prove that $$\displaystyle \int_R \sin^{n-2}\phi_1 \sin^{n-3}\phi_2\cdots\sin \phi_{n-2} d\theta d\phi_1\cdots d\phi_{n-2} = \frac{2\pi^{n/2}}{\Gamma(n/2)}$$ where $ ...
4
votes
4answers
59 views

Having trouble solving $\int\frac{5x^2+3x+2}{x(x+1)^2}$

I've first transformed the integral to $$\int\frac{5x^2+3x+2}{x(x^2+2x+1)}dx$$ Which gave me $$\frac{A}{x}+\frac{Bx+C}{x^2+2x+1}$$ $$=\frac{A(x^2+2x+1)+Bx^2+Cx}{x(x^2+2x+1)}$$ ...
4
votes
3answers
172 views

Integral fraction of polynomials

I have this problem: $$\int \frac{-2x^2+6x+8}{x^4-4x+3}dx$$ I have tried using partial fractions, but I can't get solution. Thank you for any advice.
4
votes
5answers
187 views

Solution of $\int \frac{1}{x^2 \sqrt{x^2+9}}dx$

I'm new of the site. I must solve this exercise: $$\int \frac{1}{x^2 \sqrt{x^2+9}}\,dx$$ I tried every substitution, but I didn't reach that I want. Can you help me, please?
4
votes
2answers
33 views

$f(x) = \int_0^x\frac{1-t^2}{\sqrt{t^4+1}}dt$ find it's derivative and tangent where x = 0

I am given this function: $$f(x) = \int_0^x\frac{1-t^2}{\sqrt{t^4+1}}dt$$ I have to find it's derivative $f'(x)$ and I have to find the equation of it's tangent in the point $x = 0$. I'm a bit ...
4
votes
3answers
147 views

Evaluation of integral $\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$

I'm trying to evaluate the following integral: $$\mathcal{J}=\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$$ Well there are $3$ poles , one lying on the real line the other on ...
4
votes
3answers
147 views

Dirichlet's integral $\int_{V}\ x^{p}\,y^{q}\,z^{r}\ \left(\, 1 - x - y - z\,\right)^{\,s}\,{\rm d}x\,{\rm d}y\,{\rm d}z$

I found such an exercise: Calculate the Dirichlet's integral: $$ \int_{V}\ x^{p}\,y^{q}\,z^{r}\ \left(\, 1 - x - y - z\,\right)^{\,s}\,{\rm d}x\,{\rm d}y\,{\rm d}z \quad\mbox{where}\quad p, q, r, s ...
4
votes
3answers
83 views

How find $a,b$ if $\int_{0}^{1}\frac{x^{n-1}}{1+x}dx=\frac{a}{n}+\frac{b}{n^2}+o(\frac{1}{n^2}),n\to \infty$

let $$\int_{0}^{1}\dfrac{x^{n-1}}{1+x}dx=\dfrac{a}{n}+\dfrac{b}{n^2}+o(\dfrac{1}{n^2}),n\to \infty$$ Find the $a,b$ $$\dfrac{x^{n-1}}{1+x}=x^{n-1}(1-x+x^2-x^3+\cdots)=x^{n-1}-x^n+\cdots$$ so ...
4
votes
5answers
123 views

How to integrate $\int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x$?

I encountered this integral in the quantum field theory calculation. Can I do this: $$ \left. \int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x =x\ln\left(\, x\,\right)\right\vert_{0}^{1} ...
4
votes
1answer
209 views

The proper and easiest way of doing an integral with derivative?

I have this integral: $$\int{\sec^3x\,\mathrm dx}$$ I don't understand how I would solve this. Google and YouTube videos don't help me understand much, other than just giving the answer. Is it ...
4
votes
2answers
120 views

intregration without substitution of $x^x \ln x$

How do i integrate this without any substitution, purely algebraically : $$x^x \ln ex$$ I've tried a lot but not have been able to: $$x^x \ (ln x + 1) = \ln x^{x^x} + x^x$$ or $e^{x \ln x}\ln ...
4
votes
2answers
116 views

Integrating by parts - question on a limit

I want to integrate this function by parts: $$\int_0^\infty x\lambda e^{-\lambda x} \, \mathrm{d}x$$ And I arrive to the following expression: $$\int_0^\infty x\lambda e^{-\lambda x} \, ...