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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General Integral Formula

I know how to find the integral below, but I would like to know if there is any clever or general formula for the integral, since my method involves simple polynomial division... $\int \frac{1}{1+\sqrt[n]x}dx$ Thanks.
user191919
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$ \int \frac{x^3}{\sqrt{x^2+x}}\, dx$

I'm trying to solve this irrational integral $$ \int \frac{x^3}{\sqrt{x^2+x}}\, dx$$ doing the substitution $$ x= \frac{t^2}{1-2 t}$$ according to the rule. So the integral becomes: $$ \int \frac{-2t^6}{(1-2t)^4}\, dt= \int…
Anne
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normalization factor for restricted density

Bounty update: this can be solved by change of basis, but I'm intrigued by David's solution relying on Fourier Transform of Dirac Delta function, so the bounty is for whoever finds a way to fix his solution to give the right result. Suppose I have a…
Yaroslav Bulatov
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Can we integrate over a summation index?

So I am reading this paper https://arxiv.org/pdf/math/0008177.pdf by Jeffrey Lagarias and in the proof of Lemma 3.1 he says \begin{equation} \int_{1}^{n} \frac{\lfloor t \rfloor}{t^2}dt = \sum_{1 \le r \le n} \int_{r}^{n} \frac{1}{t^2}…
Holly Morph
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Evaluate $\int_{0}^{\pi/2}\cos^{2n}(x)\text{d}x$.

I try this. Notice that, $$ \begin{split} \cos^{2n}x &= \left(\frac{e^{ix}+e^{-ix}}{2}\right)^{2n} = \frac{1}{2^{2n}} \sum_{k=0}^{2n} \binom{2n}{k}e^{ikx}e^{-i(2n-k)x} \\ &= \frac{1}{2^{2n}} \sum_{k=0}^{2n} \binom{2n}{k}e^{i(2k-2n)x}…
James A.
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Applications of Double/Triple Integrals

This is the question that I need to solve using mathematica: The concentration of an air pollutant at a point $(x,y,z)$ is given by: $$p(x,y,z) = x^2y^4z^3 \text{ particles}/m^3$$ We're interested in studying the air quality in a region in 3-space…
duxrule
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How to solve $\int_{0}^{\frac{\pi}{2}} \frac{2304\cos x}{(\cos 4x-8\cos 2x+15)^2} \,dx$

\begin{equation} \int_{0}^{\frac{\pi}{2}} \frac{2304\cos x}{(\cos 4x-8\cos 2x+15)^2} \,dx \end{equation} This is a MCQ question and there are 5 options to choose which are "A.$2\sqrt{3}\pi+9\ln 3$, B.$2\sqrt{7}\pi+8\ln 3$ , C.$2\sqrt{3}\pi+8\ln 3$…
user635988
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Integral of fractional expression $\int^3_0 \frac{dx}{1+\sqrt{x+1}}$

I want to solve this integral and think about call $\sqrt{x+1} = t \rightarrow t^2 = x+1$ $$\int^3_0 \frac{dx}{1+\sqrt{x+1}}$$ Now the integral is : $$\int^3_0 \frac{2tdt}{1+t}$$ now I need your suggestions. Thanks.
Ofir Attia
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How to do this integration?

I'm trying to integrate $$\frac{8x^2+3x+1}{x(2x+1)^2}$$ I did a partial fraction expansion: $$\frac{8x^2+3x+1}{x(2x+1)^2}= \frac{1}{x}+\frac{2}{2x+1}-\frac{3}{(2x+1)^2}$$ and now I'm left with…
Archy Will He 何魏奇
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Is Integral is considered defined even when one method gives defined results and other undefined results

something that I found confusing me. Lets see for example the follow integral: $$\int_{A}^{B}e^{t\cdot x} \cdot dx=\left[\frac{e^{t\cdot x}}{t}\right]_{A}^B=\frac{e^{t\cdot B}-e^{t\cdot A}}{t}$$ From this results we may conclude that for $t=0$ ,…
audi02
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Differentiability of an integral accumulation function

Is $$H(x) = \int_0^x \left\lvert\sin\left(\frac{1}t\right)\right\rvert\,\mathrm dt$$ differentiable at $x = 0$? I claim that $H(x)$ is differentiable at $x=0.$ Observe that \begin{align}H(-x) &=…
user747911
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What can using the "opposite" combination for integration by parts be used for

For example, $$I=\int xe^{x}dx$$ By taking the derivative of x, and then repeating integration by parts once, the integral can be evaluated trivially. However, when taking the derivative of $e^{x}$ and integrating $x^2$, the process goes on…
jamie
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Calculate:$\int \frac{px^{p+2q-1} - qx^{q-1}}{x^{2p+2q}+2x^{p+q}+1} dx $

Find following integration $$\int \frac{px^{p+2q-1} - qx^{q-1}}{x^{2p+2q}+2x^{p+q}+1} dx $$
kalpeshmpopat
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$\int_0^1 \frac{x^{p}}{x^{p+1}+(1-x)^{p+1}} dx=?$

$$\int_0^1 \frac{x^{p}}{x^{p+1}+(1-x)^{p+1}} dx=?$$ I tried to use $$\int_0^1 \frac{x^{p+1}}{x^{p+1}+(1-x)^{p+1}} dx=\frac{1}{2}$$ and integration by parts. I do not know if there is any restriction on p.in original question p=2014,Question from…
Kian
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Find:$\int_{a}^{b}\ [x] dx + \int_{a}^{b}\ [-x] dx $

Find following integration $\int_{a}^{b}\ [x] dx + \int_{a}^{b}\ [-x] dx $ where [.] denotes greatest integer function
kalpeshmpopat
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