Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

Integration is a major part of .

There are two main kinds of integrals:

  • definite integrals (e.g. proper and improper integrals), which often have numerical values
  • indefinite integrals, which group families of functions with the same derivative.

Several techniques to solve integrals have been developed, including integration by parts, substitution, trigonometric substitution, and partial fractions.

Integration can be used to find the area under a graph and find the average of the function. Also, it can be used to compute the volume of certain solids and to find the displacement of a particle.

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Integral equations problem

Show that there is no solution to: $$v(x)=1+3 \int_0^1 xy v(y) dy$$ When the solution $v(x)$ must be of the form $1+cx$ for some constant $c$. I tried solving this I am getting $x=0$. What am I doing wrong?
kiwifruit
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Evaluate the integral. $\int sin^3 (5x) dx$

The bounds are from $(\pi/5)$ to $0$. I know we use a pythagorean identity for this. So $u=\sin^2 (x)$ and $du = \sin (2x) dx$. But I'm helping trouble solving this problem.
Mahina
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How could this Integration be solved? Any possible Steps.

$$ \int \frac{\cos x + x\cdot \sin x}{x\cdot (x + \cos x)}dx $$ Any steps leading to a simple answer would be really appreciated. Thanks !
Jishan
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Integration by Parametric Integration.

I have a question that I have no idea how to start: $$\int_{0}^{1} \frac{x-1}{\ln x}dx$$ So in one of my classes, I learned a technique called parametric integration. But I have no idea how to use the technique with this question. Any hints/help…
user133458
  • 534
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Evaluating $\int_{0}^{\infty}\frac{1-e^{-t}}{t}\sin{t}\operatorname d\!t$

Find this integral $$I=\int_{0}^{\infty}\dfrac{1-e^{-t}}{t}\sin{t}\operatorname d\!t$$ I know this $$\int_{0}^{\infty}\dfrac{\sin{t}}{t}\operatorname d\!t=\dfrac{\pi}{2}$$But I can't find this value,Thank you
math110
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surface area of the part of the circular paraboloid

Find the surface area of the part of the circular paraboloid that lies inside the cylinder
user131040
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Computing $\int \dfrac{x^4}{x^3-8} dx$

I am currently stuck with one integration and I don't know what to do know. I would appreciate detailed answer, but will be happy for any :-) I am having: $$ \int \dfrac{x^4}{x^3-8} \,dx$$ I am having 2 complex roots and I am not able to find any…
Jan Vorisek
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Is it possible to find elementary integral of $\frac{\arcsin x}{x}$?

How could I possibly indefinite integral: $$ \int{\arcsin\left(x\right) \over x}\,{\rm d}x $$ using elementary functions? If it is impossible to integrate using elementary functions, Is there any way to find the definite integral in the…
Mike
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help me to find a volume of the ring shaped solid

A cylindrical drill with radius 5 is used to bore a hole through the center of a sphere of radius 8. Find the volume of the ring shaped solid that remains. Alright, my thing is that i did not understand how to set the integral
user131040
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evaluate the integral by using polar coordinate

Using polar coordinates, evaluate the integral $$ \int\int_R\sin(x^2 + y^2)dA $$ where R is the region $1\le x^2 + y^2\le 64$
user131040
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Why is this integral undefined

I was working on a math assignment and got this problem. $$\int^\infty_{-\infty}\frac{5x}{1+x^2}dx$$ I know the indefinte integral is $$\frac{-5}{2}\ln|1+x^2|+C$$ Why is it when I looked it up I was told undefined? Working with the process improper…
wolfcall
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Integrate $\frac{x^2-1}{x^2+1}\frac{1}{\sqrt{1+x^4}}dx$

An antiderivative from Spivak $$\int \frac{x^2-1}{x^2+1}\frac{1}{\sqrt{1+x^4}}dx$$ The idea I had was to write the first factor as $\left(1-\dfrac{2}{x^2+1}\right)$, but I don't see how that's helping!
Eric Auld
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Question about integral

I want to prove the inversion theorem on $R^{k}$.Then I need to compute the integral : $\int_{R^{1}}\exp^{-\sqrt{x^{2}+a^{2}}+itx}dx$ I have no ideal how to deal with it.I will appreciate your help.
gilliatt
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Stationary Phase approximtion for this type of integral

I would like to approximate the following integral for large $t$ : $I(t)=\int_0^{\pi}dx f(x)e^{iS(x)t}$ $S$ is real and $S'(0)=S'(\pi)=0$. $f$ is real and $f(0)=f(\pi)=0$ and $f'(0)=f'(\pi)=0$. Can one do this for this type of integral? Thank you.
lagoa
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Integration by parts of $\int xf(ax)f(bx) dx$

How does one integrate $\int xf(ax)f(bx) dx$? I think it cries out for integration by parts, but I don't know how to split the integrand. Thanks.
Eddie
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