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.

73636 questions
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Find volume of body bounded by $\left(\frac{x^2}{a^2} + \frac{y^2}{b^2}\right)^2 + \frac{z^4}{c^4} = \frac{z}{k}$

Find the volume of body bounded by $$\left(\frac{x^2}{a^2} + \frac{y^2}{b^2}\right)^2 + \frac{z^4}{c^4} = \frac{z}{k}.$$ What substitution is better in this case?
Desh
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Prove $\int_a^b f(x)f''(x) dx\leq0$ if $a

Let $a
user30523
  • 1,681
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Minimum value of an integral.

Suppose we have a function, $\phi:[a,b]\to\mathbb{R}_{+}$. I am trying to prove that the function: $$g(\alpha)=\int^{b}_{a}(x-\alpha)^{2}\phi(x)dx$$ attains its minimum value on $(a,b)$, and find the point in which it reaches that value (I believe…
Johnny Westerling
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Existence of sequence for an integral function in $\mathbf{R}$.

Is it true that for a function $f\in L^1({\mathbf R})$, there exists a sequence $\,x_n\rightarrow\infty$ such that $x_nf(x_n)\rightarrow 0$?
user43014
  • 301
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Why don't all odd functions integrate to $0$ from $-\infty$ to $\infty$?

Why don't all odd functions integrate to $0$ from $-\infty$ to $\infty$? As for any odd function $f(x)$, $$\int_{-a}^{a} f(x) dx = 0$$ I actually ran into trouble in a recent examination where I was asked to compute the mean and standard deviation…
Shraddheya Shendre
  • 2,431
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How to evaluate this definite integral and find limit when upper limit of definite integral tends to infinity

Find $$ \lim_{x\to\infty} x^2\int_0^x e^{t^3-x^3}\,dt $$ A. $1/3$ B. $2$ C. $\infty$ D. $2/3$ I tried to integrate the definite integral first directly in hope of putting limits afterwards but failed. I used integration by parts twice…
Matt
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Born Approximation of Polarization Potential

Potentially more math than physics here, but I'm looking for the scattering amplitude of the potential: $$U(r) = \frac{U_o}{(r^2+d^2)^2}.$$ Where $d$ is a constant. Using the first Born approximation for a spherically symmetric potential leads…
FORTRAN90
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How to solve $\int\tan^3(x)\,dx$?

How to solve $\int\tan^3(x)\,dx$ ? Do I use substitution?
xiamx
  • 449
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Calculate definite Integral

I=$\int_{-\infty}^{\infty}e^{-ax^2}dx=\sqrt\frac{\pi}{a}$ But if we change variable: $ax^2=t$ I=$\int_{0}^{\infty}\frac{e^{-t}}{2\sqrt{at}}dt=\frac{1}{2\sqrt{a}}\Gamma(\frac12)=\frac12\sqrt\frac{\pi}{a}$ Where's wrong? True answer is…
aaron koli
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How to deduce trigonometric formulae like $2 \cos(\theta)^{2}=\cos(2\theta) +1$?

Very important in integrating things like $\int \cos^{2}(\theta) d\theta$ but it is hard for me to remember them. So how do you deduce this type of formulae? If I can remember right, there was some $e^{\theta i}=\cos(\theta)+i \sin(\theta)$ trick…
hhh
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How to calculate the energy and average power of step and ramp functions?

Using a plotted function, I've come up with a function consisting of step and ramp functions; say $x(t) = 50u(t+30)+20u(t-30)+5r(t-30)$, where $u(t)$ is the unit step function and $r(t)$ is the ramp function (which I believe is just $tu(t)$). I'm…
Yuerno
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How do I integrate this (or is there a solution in a table)?

How do I integrate: $$\int \frac{1}{p+q(x-r)^2}\frac{1}{\sqrt{s+t x^2}}\, dx$$ All variables other than $x$ can be assumed to be greater than $0$ and independent of $x$. Pointers to a formula from an integration table are also sufficient. This…
user39836
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How to solve $\int\tfrac{\sqrt{x^2+1}}{(x^2+a)^{3/2}}dx$?

I am trying to find a solution for the following integration for a physics problem. Could anyone give hint on how to do this? $$ \int\frac{\sqrt{x^2+1}}{\left(x^2+a\right)^\frac{3}{2}}dx $$ Thanks for your time.
Thanushan
  • 139
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How to calculate the value of this integral?

$$ \int_{a}^{b}\frac{dx}{\sqrt{c-x^{2}(1-d\, x)}} $$ Any help is appreciated. Thanks in advance.
Norm
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In an inequality of integrals, can I multiply both integrands with the same non-negative, monotonous function?

Assume that you have two functions $f^-,f^+ :(\mathbb R^+)^n \rightarrow \mathbb R$. Also assume that every integral I use converges. Assume that we have the following inequality for every $(a_1,...,a_n) \in (\mathbb R^+)^n$:…
modnar
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