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 $\int_{-\infty}^{\infty} \frac{1}{\sqrt{x^2+a^2}}\,dx$

$$\int_{-\infty}^{\infty} \frac{1}{\sqrt{x^2+a^2}}\,dx$$ I subbed in $x=a \tan\theta$ and ended up with $\ln|\sec\theta+\tan\theta|$. Is this correct? Thanks.
user140591
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Compute $\int_{-\pi}^\pi{x^2\sin\frac{x}{2}dx}$

I have to solve this integral: $$\int_{-\pi}^\pi{x^2\sin\frac{x}{2}\mathrm dx}$$ I'm thinking about integration by parts, but I'm not sure how to either derivate or integrate the $\sin\frac{x}{2}$ part. These are solutions to that…
peroxy
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Is My Solution on Integration by Parts Correct?

For $x>0$ let $\ f(x) = \int_0^\infty e^{-t-x^2⁄t} t^{-1/2}dt $ the question wants us to show that $\ f(x) = x \int_0^\infty e^{-t-x^2⁄t} t^{-3/2}dt $ by using substitution. However I do not think any substitution works here. What I have done so…
rose
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How find $a,b$ if $\int_{0}^{1}\frac{x^{n-1}}{1+x}dx=\frac{a}{n}+\frac{b}{n^2}+o(\frac{1}{n^2}),n\to \infty$

let $$\int_{0}^{1}\dfrac{x^{n-1}}{1+x}dx=\dfrac{a}{n}+\dfrac{b}{n^2}+o(\dfrac{1}{n^2}),n\to \infty$$ Find the…
math110
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Triple integral of $|z|$

I have to calculate the $$\int_A |z| \,dx dy dz $$ with $A=\{(x,y,z): x^2+y^2+z^2\le4, x^2+y^2-2y\le0\}$. Do I use cylindrical or spherical coordinates?
Giulia B
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Odd Function Integral?

I know if $f$ is an odd function then $$\int_{-L}^L f(x)\:dx = 0$$ my question is, is the converse necessarily true? Intuitively, I feel it should be that by assuming that the integral with those symmetric bounds is zero I can somehow show $f(-x) =…
Eddie
  • 201
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Integral of $e^{\frac{y}{x}}$

How can we evaluate the following? $$\int e^{\frac{y}{x}}\ \mathrm dy$$ An explanation of the answer would be helpful. The answer I got is $ x e^{y/x}$. But not sure about the steps used for obtaining the answer...
jay
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How to calculate $\int_0^{\infty}\frac{\ln(x)}{1+x^2}\ \mathrm dx$

How does one go about calculating : $$\int_0^{\infty}\frac{\ln x}{1+x^2}dx$$ I've tried Integration by parts, and failed over and over again
M.S.E
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How to integrate $\displaystyle\int e^{\frac{x^3}{3}} \mathrm{d}x$

While solving $y'=x^2-e^y$ I'm stuck on the last step that requires to evaluate this integral. $$\displaystyle\int e^{\frac{x^3}{3}} \mathrm{d}x$$ I don't know how to approach it. I know that it will result in incomplete Gamma function along with…
UserX
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Wolfram Alpha can't solve this integral analytically

Wolfram Alpha isn't able to calculate this integral (I don't have mathematica, but I have Wolfram Pro). $$\int_{0}^{a} \frac{1}{\sqrt{(x-a)^2+(x-b)^2}} \ dx \ \ \ , \ b>a$$ This is for a physics problem. I'd appreciate either a solution or the…
Zane Dufour
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Integrating $\int \sec^2(x) \tan(x) dx$ by trig substitution

I know I am supposed to integrate $$\int \sec^2(x) \tan(x) dx$$ by substituting $u = \tan(x)$ and get $du = \sec^2(x)$. However, why can't I use $u = \sec(x)$, $du = \tan(x) \sec(x)$?
bodygued
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Help for solving $ \int \frac{e^{ax^2+bx}}{\sqrt{1-x^2}}dx $

Salam, I would appreciate it if anyone could help me solving this integral: $$ \int \frac{e^{ax^2+bx}}{\sqrt{1-x^2}}dx $$ Many thanks.
zohreh s
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General formula of $I_{2n} = \int_{-\infty}^{+\infty} e^{-x^{2n}}dx$

$I_{2n} = \int_{-\infty}^{+\infty} e^{-x^{2n}}dx$ We know very well $I_{2}= \sqrt{\pi}$ Could you please help me find general formula of $I_{2n}$? Thanks for answers
Mathlover
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Are there functions for which the cyclic integration-by-parts technique does not work?

There are a lot of functions where you can use what my teacher has described as the 'cyclic' method of integration. An example is $$\int e^x\sin x\,dx$$ where you designate $u=\sin x$ and $dv=e^x\,dx$. You do integration by parts and arrive…
HDE 226868
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