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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2answers
98 views

Finding $\int\frac{\sqrt{1-t^2}}{1+t^2}dt$

I wanted to find $\int\frac{\sqrt{1-t^2}}{1+t^2}dt$, so I substituted $t=\sin\theta$ and got $\int\frac{\cos^2\theta}{1+\sin^2\theta}d\theta$; but I'm not sure what the best way to proceed from here ...
4
votes
3answers
118 views

Proof of this definite integral?

Saw this sometime in my calculus book, from the Putnam Math Challenges listed: $$\lim _{ n\rightarrow \infty }{ \int _{ 0 }^{ 1 }{ \int _{ 0 }^{ 1 }{ \underbrace{\dots}_{n-3 \, times} \int _{ 0 }^{ ...
4
votes
3answers
80 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
1answer
75 views

Integral Inequality $\int\limits_0^1f^2(x)dx\geq12\left( \int\limits_0^1xf(x)dx\right)^2.$

Let $f:[0,1]\to\mathbb{R}$ be a continuous function such that $\int\limits_0^1f(x)dx=0$. Prove that $$\int\limits_0^1f^2(x)dx\geq12\left( \int\limits_0^1xf(x)dx\right)^2.$$ My approach as follow Let ...
4
votes
3answers
140 views

Integration by substitution notation question

Often with integration by substitution I see (and use) the notation $ x \to \frac{\pi}{2} - x $, for the simple reason that I don't have to rename the variable that I am integrating with respect to, ...
4
votes
5answers
100 views

Prove that volume of a sphere with radius $r$ is $V=\frac43r^3\pi$

Prove that volume of a sphere with radius $r$ is $V=\frac43r^3\pi$. I know to prove this in following way: If I rotate graph of the function $y=\sqrt{r^2-x^2}$ around $x$-axis, it will result a ...
4
votes
2answers
200 views

How to figure out the limit of this question

I am trying to solve this question: Suppose $X$ and $Y$ are random variables with joint density $$ f_{X,Y}(x,y) = \left\{ \begin{array} {cc} 2 &\text{ for } 0<x<y<1 \\ 0 ...
4
votes
2answers
163 views

Integration help - question: $e^{-\sin(x)}$

I would really like some help with the integration of $e^{-\sin(x)}$. Thanks to anyone who will help :) Given that $\sin(x) > \frac{2x}{\pi}$ for $0 < x < \frac{\pi}{2}$, where ...
4
votes
2answers
88 views

Why does $\int_0^1 \frac 1 { \sqrt{ x (1 - x) } } \, \mathrm d x = \pi$?

I was wondering why the following is true: $$\int_0^1 \frac 1 { \sqrt{ x (1 - x) } } \, \mathrm d x = \pi$$ It is easy to obtain this result by doing a trig substitution but it's messy and not ...
4
votes
3answers
172 views

$\sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)^3}$ using complex analysis

Evaluate: $$S = \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)^3} \space \text{using complex analysis}$$ This my question: we need to consider a $f(z)$ such that, $$\frac{1}{2\pi i} \cdot\oint_{C_N} f(z) ...
4
votes
5answers
136 views

Prove $(b-a)\cdot f(\frac{a+b}{2})\le \int_{a}^{b}f(x)dx$

Let $f$ be continuously differentiable on $[a,b]$. If $f$ is concave up, prove that $$(b-a)\cdot f\left(\frac{a+b}{2}\right)\le \int_{a}^{b}f(x)dx.$$ I know that (and have proved) $$(b-a)\cdot ...
4
votes
3answers
143 views

how to integrate $\;I(a) = \int_0^1 \frac{\ln(1-a^2x^2)}{ x^2\sqrt{1-x^2}}dx$

How to solve this integral? $$I(a) = \int_0^1 \frac{\ln(1-a^2x^2)}{x^2\sqrt{1-x^2}}\,dx$$
4
votes
4answers
1k views

Why we use dummy variables in integral?

I want to know why we use dummy variables in integral? thanks so much.
4
votes
1answer
104 views

Stuck on this intergral $\int^\frac{\pi}{3}_\frac{\pi}{4} \frac{\tan^2x}{x-\tan x} dx $ calculus I

$$\int^{\pi/3}_{\pi/4} \frac{\tan^2x}{x-\tan x} dx $$ this is that I have tried $$\int^{\pi/3}_{\pi/4} \frac{\frac{\sin^2x}{\cos^2 x}}{x-\frac{\sin x}{\cos x}} dx $$ $$\int^{\pi/3}_{\pi/4} ...
4
votes
2answers
157 views

$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx$ and $\int_{0}^{\infty} \frac {\ln(x)}{x^2+b^2} dx$

Prove that $$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx = \frac {\pi}{2e}$$ My approach would be $$\lim_{n \to \infty} \int_{0}^{n} \frac{\cos(x)}{1+x^2} dx$$ and evaluate the limits of the sine and ...
4
votes
3answers
334 views

Integral of inverse of square root of quartic function with real roots

I was doing a physics problem and in order to finish it, I need to prove that: $\int_{x1}^{x2}\frac{dx}{{((x1 - x)(x - x2)(x - x3)(x - x4))}^{1/2}} = \int_{x3}^{x4}\frac{dx}{{((x1 - x)(x - x2)(x - ...
4
votes
2answers
78 views

$\int \sqrt{1+\sin ^2 x} dx$ an elliptic integral?

It seems to be an elliptic integral of the second kind, but when $k=i$? This is going by the definition that $E(\theta,k)=\int_{0}^{\theta} \sqrt{1-k^2 \sin^2x}dx$. That seems a bit off. Or is this ...
4
votes
2answers
149 views

Prove: $\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3$

I'd like to evaluate the following definite integral: $$\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3.$$ ...
4
votes
2answers
910 views

Comparison of Newton-Cotes Quadrature and Gaussian Quadrature formulas

Newton-Cotes quadrature formulas are a generalization of trapezoidal and Simpson's rule. The trapezoidal rule involves $2$ points, Simpson's rule involves $3$, and in general Newton-Cotes formulas ...
4
votes
1answer
103 views

Integrate : $\int(\sin x+\cos x)^ndx$

Problem : $$\int(\sin x+\cos x)^n\ dx$$ I am not getting any clue how to integrate this. Please help . I will be grateful to you. Thanks.
4
votes
2answers
131 views

Letting $N\to\infty$ in an integral containing $\sin Nq$

I'm stuck in a path on a paper about thermal conductivity. There is a identity involving an integral that a I can't realize how they've perfomed it. Here is it: $$\lim_{N\to \infty} ...
4
votes
4answers
79 views

Integral can't find how to do it: $\int\frac{2\ln(x)}{x}dx$

I have to find this integral $$\int\frac{2\ln(x)}{x}dx$$ This is how I began: $$\int\frac{2\ln(x)}{x}dx=2\int\frac{\ln(x)}{x}dx$$ Then I tried substitution $e^u=x$ to get $u=\ln(x)\longrightarrow ...
4
votes
4answers
145 views

How to integrate $\int \frac{\cos x}{\sqrt{\sin2x}} \,dx$?

How to integrate $$\int \frac{\cos x}{\sqrt{\sin2x}} \,dx$$ ? I have: $$\int \frac{\cos x}{\sqrt{\sin2x}} \,dx = \int \frac{\cos x}{\sqrt{2\sin x\cos x}} \,dx = \frac{1}{\sqrt2}\int \frac{\cos ...
4
votes
4answers
198 views

Integral $P\int_0^\infty \frac{x^{\lambda-1}}{1-x} dx$

I am trying to calculate the following principle value integral \begin{equation} P\int_0^\infty \frac{x^{\lambda-1}}{1-x} dx \end{equation} for $\lambda \in [0,1].$ I tried to turn this into a ...
4
votes
4answers
445 views

Proof of Integration formula

$$\int_0^{\infty}x^{-1}e^{-ax}\sin (bx) \;\mathrm dx = \arctan \frac{b}{a}$$ How to prove this result?
4
votes
5answers
329 views

Integral, definite integral

How can we prove $$ \int_0^1 \frac{\ln x \cdot \ln(1+x)}{1+x}dx=-\frac{\zeta(3)}{8}? $$ This has been one of the integrals that came out of an integral from another post on here, but no solution to ...
4
votes
4answers
151 views

How find this integral $F(y)=\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)(1+(x+y)^2)}$

Find this integral $$F(y)=\int_{-\infty}^{\infty}\dfrac{dx}{(1+x^2)(1+(x+y)^2)}$$ my try: since $$F(-y)=\int_{-\infty}^{\infty}\dfrac{dx}{(1+x^2)(1+(x-y)^2)}$$ let $x=-u$,then ...
4
votes
2answers
118 views

Proof of integral

Is there an analytical method to show that $$ \int_{-a}^a\exp\left(\frac{-1}{1-(x/a)^2}\right)\,\mathrm{d}x=ka, $$ for $a>0$. I have confirmed this result numerically for a range of values of $a$. ...
4
votes
1answer
2k 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 ...
4
votes
2answers
346 views

How to solve this integral: $\int_{-1}^{1} x^k (1-x^2)^{(n/2)-2} \, dx$

How to solve this integral step by step: $$\int_{-1}^{1} (x^k) (1-x^2)^{(n/2)-2}dx=??? $$ In my text book, it shows the result like below: $$\int_{-1}^{1} (x^k) (1-x^2)^{(n/2)-2}dx= ...
4
votes
3answers
3k views

How to master integration and derivation?

We have learnt in school about derivation and integration, however I find my knowledge fairly poor. I mean I have problems with taking the derivative/integrating even simple functions. So I would like ...
4
votes
1answer
84 views

How to prove that $\int_0^{\pi} \log(|\sin t|)\,\textrm{dt} \;\;\textrm{is integrable }$

How to prove that $$\int_0^{\pi} \log(|\sin t|)\,\textrm{dt} \;\;\textrm{is integrable }$$ Any hints would be appreciated.
4
votes
2answers
164 views

How to evaluate $\int_{0}^{\pi} \log(2+\cos x)dx$? [duplicate]

I tried integrating it by part but it didn't work. I konw it can be solved by the Gauss Mean Value theorem .Is there some elementary method for evaluate it or just some others? It seems there are at ...
4
votes
4answers
76 views

Integrating $\int \frac{x^2+1}{x(x^2-1)}$

How would I integrate the following. $$\int \frac{x^2+1}{x(x^2-1)}$$ I have done the following. $$\frac{x^2}{(x)(x+1)(x-1)}$$ $$\frac{A}{x}+\frac{B}{x+1}+\frac{C}{x-1}$$ I then did $\quad ...
4
votes
3answers
356 views

Two functions agreeing except on set of measure zero

Let $f,g:S\rightarrow\mathbb{R}$; assume $f$ and $g$ are integrable over $S$. Show that if $f$ and $g$ agree except on a set of measure zero, then $\int_Sf=\int_Sg$. Since $f$ and $g$ are ...
4
votes
1answer
187 views

Prove that $\int_{0}^{1}|f(x)f'(x)|dx\leq\frac{1}{2}\int_{0}^{1}|f'(x)|^2dx$

Let $f$ be a continuously differentiable function on $[0,1]$ and $f(0)=0$. Prove that $$\int_{0}^{1}|f(x)f'(x)|dx\leq\frac{1}{2}\int_{0}^{1}|f'(x)|^2dx$$ Thank you!
4
votes
2answers
136 views

Integration involving fixed points

A few days ago I ran into this statement If $a$, $b$ are fixed points of a function $f$, then $$\int_a^b(f(x) + f^{-1}(x)) \,\mathrm dx = b^2 - a^2.$$ I checked it for a few simple cases like ...
4
votes
3answers
143 views

An identity involving the Beta function

I'm trying to show that $$ \int _0^1 \frac{x^{a-1}(1-x)^{b-1}}{(x+c)^{a+b}}dx = \frac{B(a,b)}{(1+c)^ac^b}$$ Where $$B(a,b) := \int _0^1 x^{a-1}(1-x)^{b-1}dx $$ is the "Beta function". I am ...
4
votes
2answers
98 views

Inequality with Gamma function: how to prove it?

Let $\alpha \in (0,1)$ and $\Gamma(\alpha) = \int_0^{\infty}s^{\alpha - 1}e^{-s}ds$. I would like to prove that $$\int_0^{\infty}\frac{s^{-\alpha}}{1 + s}ds \le \Gamma(1 - \alpha)\Gamma(\alpha).$$ ...
4
votes
3answers
3k views

Algebraic way to change limits of integration of a double integral

I know how to graphically change the limits of integration of a double integral. That is, by graphing the region and eyeballing (a.ka.a "looking at") it to determine the new limits. But an answer to a ...
4
votes
2answers
478 views

Integrating $\int_0^\infty \sin(1/x^2) \, \operatorname{d}\!x$

How would one compute the following improper integral: $$\int_0^\infty \sin\left(\frac{1}{x^2}\right) \, \operatorname{d}\!x$$ without any knowledge of Fresnel equations? I was thinking of using ...
4
votes
3answers
221 views

How is this linear 2nd-order ODE solved?

In this article, the authors present the inhomogeneous equation $$\ddot{\phi}_2 + \phi_2 + g_2\phi_1^2 + \omega_1\ddot{\phi}_1 = 0,\tag{11}$$ where $$ \phi_1 = p_1 \cos(\tau + \alpha), \tag{13}$$ ...
4
votes
2answers
155 views

Integration and fundamental Theorem of Calculus

I need some help with the following integration/use of fundamental theorem of calculus: $\displaystyle x(t) = \int_{0}^{t} \exp (-2s)a(s) \ ds$, where $a(x) = \left\{ \begin{array}{lr} ...
4
votes
1answer
269 views

Asymptotic expansion of the integral $\int_2^x \frac{e^t}{t} dt$ for $x \to \infty$

Hello I wonder if there is any asymptotics known for such integral: $$ I(x) = \int_2^x \frac{e^t}{t} dt \qquad\text{when $ x\to+\infty $}. $$ Thank you very much.
4
votes
2answers
529 views

How to integrate $\int x^2 \sin^2(x)dx$

I don't know how to integrate $\int x^2\sin^2(x)\,\mathrm dx$. I know that I should do it by parts, where I have $$ u=x^2\quad v'=\sin^2x \\ u'=2x \quad v={-\sin x\cos x+x\over 2}$$ and now I have ...
4
votes
3answers
958 views

Evaluate the following integral, $\int\sqrt{4-\sqrt{x}}dx$

Evaluate the following integral, $$\int\sqrt{4-\sqrt{x}}dx$$ $$\int \sqrt{4-\sqrt{x}}dx=\int \sqrt{2^2-(x^{1/4})^2}dx$$ Considering the common subsitution for $a^2-x^2$, let ...
4
votes
3answers
1k views

Proving two integral inequalities

Can anyone help me to prove that these integral inequalities hold? Here $x$ is a real value: $$ \left| \int_a^b\ f(x) dx \right| \leq \int_a^b\ |f(x)| dx $$ Here $z$ is a complex value: $$ \left| ...
4
votes
3answers
743 views

How to evaluate the integral: $\int\ln x\;\sin^{-1} x\, \operatorname d\!x$?

Can the following integral integrated by parts? $$\int\ln x\;\sin^{-1} x\, \operatorname d\!x$$
4
votes
2answers
1k views

Complex part of a contour integration not using contour integration

A propos of a user's comment on this question, quoting Feynman to the effect that some integrals are only possible using contour integration, I wonder what the simplest example of such an integral ...
4
votes
4answers
1k views

Weak and pointwise convergence in a $L^2$ space

Let $I$ be a measured space (typically an interval of $\Bbb R$ with the Lebesgue measure), and let $(f_n)_n$ a sequence of function of $L^2(I)$. Assume that the sequence $(f_n)$ converge pointwise ...