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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184 views

Computing an integral with exponent and sqrt

Can someone help me to evaluate the integral $$ \int_{b}^\infty \sqrt{x} e^{-ax}dx. $$ I guessed that it has no elementary anti-derivative, and indeed substituting $x=t^2$ and then applying ...
3
votes
2answers
89 views

Integral, u substitution

This problem is giving me a headache. $$ \int \frac {4/7 + \sqrt{x\sqrt{x}}} {\sqrt{4-x(1+x^{3/4})}} \,dx $$ I tried simplifying the x's but im still lost. $$ \int \frac {4/7 + x^{3/4}} ...
3
votes
2answers
254 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} ...
3
votes
3answers
151 views

How to do this interesting integration?

$$\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+\Delta x}^{k+1-\Delta x}x^m dx$$ How to integrate the above integral? Edit1: $$\lim_{\Delta x\rightarrow0}\int_{2-\Delta x}^{2+\Delta x}x^m ...
3
votes
2answers
243 views

When $ f(x) = \int{f(x)} $

If $ f(x) = \int_{-\infty}^x{f(t) d t} $ means $ f(x) = B e^x $ Then $ f(x) = \int_0^x{f(t) d t} $ means $ f(x)=? $ EDIT: The obvious extension: $ f_a(x) = \int_a^x{f_a(t) d t} $ means $ f_a(x)=? $
3
votes
2answers
182 views

Calculate a double integral

I would like to ask a pretty easy question (at least I believe so). I know that: $$\phi_{11}(k) = \frac{E(k)}{4\pi k^4}(k^2 - k_1^2)$$ $$E(k) = \alpha ...
3
votes
2answers
188 views

Simple integration question.

integrate $$ \int \sin(x) \cos(x)\; dx $$ using $u$-substitution. If i take $u = \sin(x)$ I get final answer to be $\sin^2(x) / 2 + c$ But If i take $u = \cos(x)$ I get final ...
3
votes
2answers
149 views

Computation of a certain integral

I would like to compute the following integral. This is for a complex analysis course but I managed to around some other integrals using real analysis methodologies. Hopefully one might be able to do ...
3
votes
2answers
173 views

Show that $(\int_0^1 f(x)~dx)^2\leq2\int_0^1x (f(x))^2~dx $

Let $f$ be a non-decreasing, integrable function defined on $[0, 1]$. Show that $$(\int_0^1 f(x)~dx)^2\leq2\int_0^1x (f(x))^2~dx $$
3
votes
1answer
257 views

How to evaluate $\int \frac{\mathrm dx}{\sqrt[3]{\tan\,x}}$?

Please show me the steps of the following integration. I got an answer in Wolfram, but I need steps.. $$\int \frac{\mathrm dx}{\sqrt[3]{\tan\,x}}$$
3
votes
4answers
182 views

Determine divergence of $\int_0^\infty\frac{ e^x}{x}$ with limit comparison test.

Today I was asked if you can determine the divergence of $$\int_0^\infty \frac{e^x}{x}dx$$ using the limit comparison test. I've tried things like $e^x$, $\frac{1}{x}$, I even tried changing bounds ...
3
votes
4answers
94 views

Integration and Limits

I suspect the following integration to be wrong. My answer is coming out to be $3/5$, but the solution says $1$. $$\int_0^1\frac{2(x+2)}{5}\,dx=\left.\frac{(x+2)^2}{5}\;\right|_0^1=1.$$ Please help ...
3
votes
2answers
196 views

Evaluating $\int \sqrt{x^2 + 2x}dx$

$$\int \sqrt{x^2 + 2x}dx$$ I have no clue what to do on this problem. It is in the trig substitution chapter so I know I have to use that somehow. I know that I can not complete the square because ...
3
votes
2answers
184 views

How can I prove $\int[F(x+a)-F(x)]\,dx=a$

How can I prove $$\int[F(x+a)-F(x)]\,dx=a$$ where $F(x)$ is the cumulative distribution function?
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votes
2answers
2k views

Integrate $\csc^3{x} \ dx$

I found these step which explain how to integrate $\csc^3{x} \ dx$. I understand everything, except the step I highlighted below. How did we go from: $$\int\frac{\csc^2 x - \csc x \cot x}{\csc x - ...
3
votes
2answers
291 views

Evaluating the definite integral $\int_{-\infty}^{+\infty} \mathrm{e}^{-x^2}x^n\,\mathrm{d}x$

I recognize that the $\int_0^\infty \mathrm{e}^{-x}x^n\,\mathrm{d}x = \Gamma(n+1)$ and $\int_{-\infty}^{+\infty} \mathrm{e}^{-x^2}\,\mathrm{d}x = \sqrt{\pi}$. I am having difficulty, however with ...
3
votes
1answer
243 views

Steps in evaluating $\int \frac{1}{x^4-c^4} dx$?

$$ \int\ \frac {dx} {x^4 - c^4} \\ $$ is equal to (from integral tables) $$ \frac {1} {4 c^3} \ln \frac {x-c} {x+c} - \frac {1} {2 c^3} \tan^{-1}\frac {x} {c}$$ If I let $\frac{x}{c}=\tan u$, then ...
3
votes
3answers
2k views

If $f$ is a Riemann integrable prove $|f|$ is also Riemann integrable

Show that if $f$ is Riemann integrable on $[a,b]$ then $|f|$ is also Riemann integrable on $[a,b]$. My idea is: let $f$ be in $[a,b]$ less than $|f|$, since $f$ is integrable then $|f|$ is also ...
3
votes
3answers
292 views

Integrate $\int\frac{1}{x^6} \sqrt{(1-x^2)^3} ~ dx$

How to integrate the following? $$\int\frac{\sqrt{(1-x^2)^3}}{x^6} \;dx .$$
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votes
2answers
1k views

Integral related to Arcsinh(x)

It is not difficult to verify that $$ \frac{\mathrm d}{\mathrm dx} \left[ \log\Big(x+\sqrt{x^2+1}\Big) \right] = \frac{1}{\sqrt{1+x^2}} $$ for $x\geq 0$, say. How would one calculate the indefinite ...
3
votes
2answers
657 views

Torricelli's/Gabriel's Trumpet Surface Area

For an assignment we have been asked to compute the surface area of Torricelli's trumpet which is obtained by revolving $y=1/x$ where $x>=1$ about the x axis. We have to calculate the surface area ...
3
votes
2answers
134 views

partial integration help, please?

if $\phi(x)={1\over x}\int_0^x F(t)dt$ and $F(x):=\int_0^x f(t)dt$ ,how does $$\phi'(x)=-{1\over x^2}\int_0^xF(t)dt +{1\over x}F(x)={1\over x^2}\int_0^x t f(t)dt\ ?$$ Please explain step by step ...
3
votes
3answers
363 views

$\int \frac{dx}{(x^4 + 1)^2}$

What would be a relatively simple method for computing the indefinite integral below? $\displaystyle \int \frac{dx}{(x^4+1)^2}$ Furthermore, how would one evaluate the following, possibly by ...
3
votes
3answers
265 views

How to find: $\int_{-d}^d \frac{z}{2\sqrt{z+c}\ \cdot \sqrt{d^2-z^2}} dz$

Let $c,d>0$. I do not know how to integrate the following: $$\int_{-d}^d \frac{z}{2\sqrt{z+c}\ \cdot \sqrt{d^2-z^2}}\ \text{d}z.$$ Sorry~~ it should be $z^2$. I think it can be simplified to ...
3
votes
2answers
132 views

Why does integration of acceleration data create a slope?

I created a 100hz sine wave in code. When I graph the waveform I get this: When I do an integration on this pure sine wave to get a velocity waveform I get: Is this normal? I do not have a ...
3
votes
2answers
94 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: $$\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3.$$ ...
3
votes
2answers
98 views

How to compute $\int \frac{x}{(x^2-4x+8)^2} \mathrm dx$?

Can someone help me to compute: $$\int \frac{x}{(x^2-4x+8)^2}\mathrm dx$$ And, in general, the type: $$\int \frac{N(x)}{(x^2+px+q)^n}\mathrm dx$$ with the order of polynomial $N(x)<n$ and $n$ ...
3
votes
3answers
75 views

Computation of $\int{\sqrt{c^2+b^2t^2}\,\mathrm dt}$.

How to solve $$\int{\sqrt{c^2+b^2t^2}\,\mathrm dt}$$ I tried substituting $t$ with $\sin{x}$ but it doesn't work, since $b^2$ creates a problem.
3
votes
1answer
61 views

$\int\limits_0^4\int\limits_\sqrt x^2 \dfrac x{1+y^5}\ dy\ dx$

How to find $$ \int_0^4\int_\sqrt x^2 \dfrac x{1+y^5}\ dy\ dx=\ ...? $$ The correct answer from my textbook is $\frac1{10}\log33$. I can answer it if the integrand is $\dfrac x{1+y^2}$ so I can let ...
3
votes
3answers
58 views

Find dimension of the intriguing vector space

We are given vector space of polynomials over $\mathbb R$ of two variables with powers not higher than 2013. Let's consider subspace $V$ which contains such polynomials $f$, so following holds for ...
3
votes
3answers
122 views

Need some advice to solve this integral $\int\frac{\sin^2x}{1+\sin^2x}\mathrm dx$

I'm trying to use this subtitution $t=\tan(x/2)$. But I don´t get anywhere. I've tried $t=\tan(x)$ too. Appreciate your help. $$\int\dfrac{\sin^2x}{1+\sin^2x}\mathrm dx$$
3
votes
2answers
174 views

Integral $\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$

Consider $$\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$$ Is it equal to $0$ ? Why ? Any hint ?
3
votes
3answers
277 views

Laplace's Equation in Spherical Coordinates

The general solution of the Laplace equation in spherical coordinates is (independant of $\phi$): $$V(r,\theta ) = \sum ^{\infty} _{l=0} \left( A_l r^l + \frac{B_l}{r^{l+1}} \right) P_l (\cos \theta ...
3
votes
5answers
70 views

Need help with single variable integral

I need help with the following integral. I used integration by parts to get started but I have no idea where to go from there. If someone could please show me step by step how to solve this, I would ...
3
votes
3answers
125 views

How to calculate $\int \frac{x}{\sqrt x -2}dx$

I don't know how to solve the following integral. I need some suggestions. Thank you! $$ \int \frac{x}{\sqrt x -2}dx$$
3
votes
2answers
93 views

Find the volume between two surfaces

Find the volume between $z=x^2$ and $z=4-x^2-y^2$ I made the plot and it looks like this: It seems that the projection over the $xy$-plane is an ellipse, because if $z=x^2$ and $z=4-x^2-y^2$ ...
3
votes
2answers
84 views

What happens when $\lvert\omega\rvert =1$?

If this is a duplicate in any way, I'm very sorry. I'm brushing up on some Complex Analysis with Special Functions in mind. Here's a problem I'm stuck on. Evaluate the integral $$I=\frac{1}{2\pi ...
3
votes
2answers
97 views

Finding $\frac{dy}{dx}$ of $y=\int \limits_{\sin x}^{\cos x}(\cos(\pi t^2)) dt$

If we have: $y=\int \limits_{\sin x}^{\cos x}(\cos(\pi t^2)) dt$ And i want to find $\dfrac{dy}{dx}$, so my question is: can i find it without calculating the integral? Because $y'=(\cos(\pi t^2))$. ...
3
votes
2answers
199 views

Any ideas to the integral $\int_0^\frac{\pi}{2}x\ln(\sin x)~dx$ ?

I've corrected typing error in the integral. I apologize for my mistake. Reedited question: Can anybody solve integral: $\int_0^\frac{\pi}{2}x\ln(\sin x)~dx$ ? I'm just trying to guess some simple ...
3
votes
2answers
158 views

Integral involving logarithm and cosine

For $a,b>0$ I would like to compute the integral $$I=\int_0^{2\pi} -\log{\sqrt{a^2+b^2-2ab\cos{t}}}~dt.$$ Numerical computations suggest that $$I=\min\{-\log{a},-\log{b}\}.$$ How can I prove this? ...
3
votes
2answers
117 views

Fast way to integrate $\frac{x^2-y^2}{(x^2+y^2)^2} dx \,dy$ in unit square

I am looking for a fast way to integrate $$ \int_0^1 \int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} dx \,dy$$ using standard techniques ( no complex analysis and no functional analysis). I am aware that ...
3
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2answers
67 views

Proof about boundedness of $\rm Si$

$\def\Si{{\rm Si}}$ I want to prove the boundedness of $$\Si(x) := \int_0^x \frac {\sin \xi} \xi d\xi$$ as part of a homework (about the non-surjectivity of $\mathcal F : L^1(\mathbb R) \to ...
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2answers
133 views

Reference for integration

Does anyone have a good reference for a book that already assumes knowledge of calculus/analysis and whose main focus is computing more difficult integrals? I'm looking for something which will have a ...
3
votes
3answers
137 views

Integrate $\int \frac{x^2}{(x^2+4)^2}$

$$\int \frac{x^2\;dx}{(x^2+4)^2}$$I suppose you must have to use trigonometric substitution, or something, but do not even know where to begin to solve this integral, guys, please help me!!
3
votes
2answers
814 views

Trig. Indefinite Integral $\int\frac{\tan x +\tan ^3 x}{1+\tan^3 x}dx$

$\displaystyle \int\frac{\tan x +\tan ^3 x}{1+\tan^3 x}dx$ $\underline{\bf{My \; Try}}$:: Let $\tan x = t$. Then $\sec^2 xdx = dt\Rightarrow \displaystyle dx = \frac{1}{1+\tan^2 t}dt\Rightarrow dx = ...
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votes
1answer
1k views

How to integrate greatest integer function $\int^{1.5}_0 \lfloor x^2 \rfloor \, dx$

How to integrate greatest integer function $$\int^{1.5}_0 \lfloor x^2\rfloor \, dx$$ I don't have any idea how to integrate greatest integer function, only have idea about the function viz. If $x ...
3
votes
4answers
660 views

Volume of cube section above intersection with plane

Suppose we have a unit cube (side=1) and a plane with equation $x+y+z=\alpha$. I'd like to compute the volume of the region that results once the plane sections the cube (above the plane). There are ...
3
votes
2answers
101 views

Double integral

Calculate the iterated integral $$\int_{1} ^4\int_{1} ^2 \left(\frac xy+\frac yx\right)\,dy\,dx$$ This is the work that I've done, but it'd lead me to the wrong answer, so either I did it completely ...
3
votes
3answers
117 views

How to evaluate the definite integral?

How to evaluate the definite integral? $$\int \frac{7}{3x+1}dx$$ I am having difficulties to finish the question: Below is what I did: $$ =\left.\frac{7}{3}\ln|3x+1|\right|_0^4$$ ...
3
votes
2answers
116 views

Evaluate $\int_2^\infty{\frac{3x-2}{x^2(x-1)}}$

To be shown that $\int_2^\infty{\dfrac{3x-2}{x^2(x-1)}}=1-\ln2$ My thought: $\dfrac{3x-2}{x^2(x-1)}=\dfrac{3x}{x^2(x-1)}-\dfrac{2}{x^2(x-1)}$ • ...