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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Integral $\int\frac{dx}{x\sqrt{x^2+4}}$

While solving…
RE60K
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Mean (average) distance of point to line segment

A line segment is defined as the two points $(x_1, y_1)$ and $(x_2, y_2)$. A point is then defined as $(x_3, y_3)$. My goal is to compute the mean (average) distance between the point and the line segment. (Not the shortest or longest distance). My…
Sirisian
  • 143
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Calculate $\int \log(1+\log(x))x^ndx$

How to calculate the following integral? $$\int \log(1+\log(x))x^ndx,$$ $n$ is an integer $\in N$.
Kira
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Integrate a division of polynomials

Hi I have the following integral: $$\int \frac{2x}{x^2+6x+3}\, dx$$ I made some changes like: $$\int \dfrac{2x+6-6}{x^2+6x+3}\, dx$$ then I have: $$\int \dfrac{2x+6}{x^2+6x+3}\, dx -\int\dfrac{6}{x^2+6x+3}\, dx$$ and thus:…
Luis Armando
  • 343
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Generalising integration by parts for the product of more than two functions

Just as the product rule can be generalised to the product of more than two functions, i.e. $$\frac{d}{dx} \left [ \prod_{i=1}^k f_i(x) \right ] = \sum_{i=1}^k \left(\frac{d}{dx} f_i(x) \prod_{j\ne i} f_j(x) \right) = \left( \prod_{i=1}^k f_i(x)…
beep-boop
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Double Integral transformation to Polar coordinates

Here's the question from an exam that I couldn't solve: If $\int_1^2 \int_0^x \frac{1}{(x^2+y^2)^\frac{3}{2}} ~\mathrm{dy} ~\mathrm{dx}$ transforms to $\int_0^a \int_b^c \frac{1}{r^2} ~\mathrm{dr} ~\mathrm{d\theta}$ in the polar coordinates $(r,…
hola
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How is $\,\int(1/x)\,dx = \ln|x|\,$ true?

Why does the integral of $\frac 1x dx$ equal the natural log of the absolute value of $x$? $$\int \frac 1x\,dx = \ln|x| + C$$
user8028
  • 431
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Definite integral $\int_0^{2\pi}(\cos^2(x)+a^2)^{-1}dx$

How do I prove the following? $$ I(a)=\int_0^{2\pi} \frac{\mathrm{d}x}{\cos^2(x)+a^2}=\frac{2\pi}{a\sqrt{a^2+1}}$$
Emerson
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what is the best way to solve this integral:

have this integral, and looking for the best\quickest way to solve it: $$\int_{-b}^bD\sin\left({\pi ny \over b}\right)\sin\left({\pi n'y \over b}\right)dy$$
user107761
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How prove or disprove this limit $\lim_{n\to\infty}\frac{\sum_{k=1}^{n}|\sin{(x+k)}|}{n}=\dfrac{1}{\pi}\int_{0}^{\pi}|\sin{x}|dx$

prove or disprove $$\lim_{n\to\infty}\dfrac{\displaystyle\sum_{k=1}^{n}|\sin{(x+k)}|}{n}=\dfrac{1}{\pi}\int_{0}^{\pi}|\sin{x}|dx$$ if this is right,and How prove it? Thank you My idea maybe this can use Uniform distribution? and I don't prove it.
math110
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Calculate $\int_{0}^{1}\dfrac{x^{2n}}{\sqrt{1-x^2}}\mathrm{d}x$

For integer $n\ge0$, Calculate: $$\int_{0}^{1}\dfrac{x^{2n}}{\sqrt{1-x^2}}\mathrm{d}x.$$ I would like to get suggestions on how to calculate it? Should I expand $(1-x^2)^{-1/2}$ as a series? Thanks.
Jika
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Calculate $\int \frac{1}{x^2+x+1}\mathrm{d}x$

Define the integral $I$ as follow: $$I=\int \dfrac{1}{x^2+x+1}\mathrm{d}x.$$ I do not know how to integrate it. Any suggestions please? I tried a lot of methods: I substituted $x^2+x=u$. I modified the denominator $x^2+x+1=(x+1)^2-x$ and I…
x.y.z...
  • 1,150
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Evaluate $\int\limits_0^\pi \frac{x}{1+\sin^2x} \ dx$

How can one evaluate $$\int_0^\pi \frac{x}{1+\sin^2x} \ dx\ ?$$
liesel
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How find this integral $I=\int_{0}^{1}\frac{1}{2-x}\ln{\frac{1}{x}}dx$

Find this integral $$I=\int_{0}^{1}{1 \over 2 - x}\,\ln\left(1 \over x\right)\,{\rm d}x$$ My idea: let $1-x=t$, then $$I=\int_{0}^{1}{\ln\left(1 - t\right) \over 1 + t}\,{\rm d}t$$
math110
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Integrate $\int_0^1 \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$

integrate $$\int_0^1 \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$$ I've started by dividing this into two integrals: $$\int_0^{1/2} \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$$ and $$\int_{1/2}^1 \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$$ Then I'm trying to find a primitive…
iveqy
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