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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Calculate: $\int_{(x^2+y^2+z^2)^2\le2xyz} 1 d\lambda_3(x,y,z)$

Calculate: $\int_{(x^2+y^2+z^2)^2\le2xyz} 1 d\lambda_3(x,y,z)$ I have a problem finding a substitution that will give me nice integration limits. I tried as well as $x=r\sin \alpha, y=\cos \alpha, z=z$ as $x=r\cos\alpha \cos \beta, y=r\cos \beta…
qerty149
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Which integration is correct? Or are both correct?

What's the correct method to integrate the function $\frac{1}{2(3x+1)}$? \begin{align} \int\frac{1}{2(3x+1)} dx &= \int\frac{1}{6x+2}dx\\ &= \frac{1}{6}ln(6x+2)+c \end{align} $$or$$ \begin{align} \int\frac{1}{2(3x+1)} dx &=…
Cheng
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How to evaluate this strange double integral (Vardi's integral type)?

I found this strange double integral from some a site (Can't remember where). I would like to know how this integral it is evalauted: $$\int_{0}^{1}\int_{0}^{1}\ln^2(x)…
Sibawayh
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Integral equation, certain rule

I have the following equation: $$2\int_{0}^{1/n}(1-nx)^{2}dx=\frac{2}{n}\int_{0}^{1}(1-x)^{2}dx.$$ My question is: Is this a rule and where does it come from and if so when are you allowed to use it? What I have done so far: I calculated both the…
Lech121
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How to find the integration of $\int \limits _{-\infty}^x e ^ \frac{-t^2}{2}{d}t$?

What is the value of the $\displaystyle \int \limits_{-\infty}^xe ^ {\large{-t^2/2}}dt$ ? thank you for your time.
prasad
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True or False: If $f>0$ is integrable on $[a,b]$ then $\sqrt{f}$ is integrable

I'm having trouble proving or finding a counterexample for the following statement: If $f>0$ is integrable on $[a,b]$ then $\sqrt{f}$ is integrable We're using the Riemann integral definition: If $f$ is integrable on $[a,b]$ then given…
Paz
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From Faraday's law in integral form to the differential form(=Maxwell equation) by using Stokes

I want to understand how Stoke's theorem shows that the integral form of Faraday's law: $$\int_{c(A)} E dr = -\frac{1}{c} \frac{d}{dt} \int_A B ds$$ ($A$ is a surface and $c(A)$ its boundary curve) is equivalent to its differential form, i.e. the…
Mekanik
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Integral $\int \arcsin \left(\sqrt{\frac{x}{1-x}}\right)dx$

$$\int \arcsin\left(\sqrt{\frac{x}{1-x}}\right)dx$$ I'm trying to solve this integral from GN Berman's Problems on a course of mathematical analysis (Question number 1845) I tried substituting $x$ for $t^2$: $$2\int…
Praneeth Jain
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Definite integral of Gaussian times $e^{-e^{-x}}$

I had the following problem $$ \int \limits_{- \infty}^\infty \mathrm{d} x \exp \left( - \frac{x^2}{2 \sigma} - 2 x \right) \exp \left( - E e^{-x} \right), \quad \sigma, E > 0 $$ by substituting $t = x - \log E$ I got rid of constants in the double…
user16320
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Calculate the length of the curve 's bow.

I have to calculate the length bow $$r=4\sin^4\left(\frac{x}4\right), \space x∈(0,\pi)$$ I know the formula that $$D= \int^\pi_0 { \sqrt{1+(r^*)^2}} dx$$ The $r^*$ means the derivative of $r$ so…
Kukoz
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Integration by parts of a normalized function - [copied from Physics.SE]

By using integration by parts, I need to show for $$A = \frac{\mathrm d}{\mathrm dx} + \tanh x, \qquad A^{\dagger} = - \frac{\mathrm d}{\mathrm dx} + \tanh x,$$ that $$\int_{-\infty}^{\infty}\psi^* A^{\dagger}A\psi\;\mathrm dx =…
user27182
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How do I solve this problem with U-substitution?

$$\int \left(4-x(16-x^2)^{1/2}\right)\,dx $$ I learned today I could use U-substitution to before integrating, which makes it easier to integrate. So I can make $U=16-x^2,\quad \dfrac{du}{dx} = -2x^2$ And I've no idea how I should proceed next.…
user70994
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How To Solve This Multiple Integral?

I have to solve this integral: $$ \int_D \frac {4x 4y 4z}{x^2}dxdydz, \quad \text{where} \quad D=\left\{(x, y,z) \in \mathbb R^3 \mid x,z \in [1,e], \ 0\le y\le\sqrt{\ln z}\right\}$$ I started with $$ \int_1^e \int_0^{\sqrt{\ln z}} \int_1^e \frac…
Eiden
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Find the value of k that bisects the area

Find the value of $k$ for which the line $ y= kx$ bisects the area enclosed by the curve $4y=4x-x^2$ and the $x$ - axis. I have tried to solve this and the solution seems odd... the solution that came out was $k=1 - \sqrt [3]{2}$ The step I took was…
MathsGoogle
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what is the indefinite integral of $\int \frac{x^n}{x+1}\ dx$ where $n \in \mathbb{N}$

I'm trying to solve $\int \frac{x^n}{x+1}\ dx$ for $n \in \mathbb{N}$. I tried a couple of things like trigonometric substitution and by parts but somehow it didn't help much.
Pawan Nirpal
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