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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Double Integration over finite plane.

$$\phi(z)=\frac{\sigma}{4\pi\varepsilon_0}\int_{\frac{-a}{2}}^{\frac{a}{2}}\int_{\frac{-a}{2}}^{\frac{a}{2}}\frac{1}{\sqrt{x^2+y^2+z^2}}~dx~dy$$ I'm not sure how to do this integral. For the first integral w.r.t $x$ i tried to substitute…
NormalsNotFar
  • 803
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Integral with trigonometric function

I have a problem with this integral $$\int_\ \frac{\sin 2x }{ \sqrt{4-\cos^2 x}} \, dx$$ We can transform it to $$\int_\ \frac{2\sin x \cos x }{ \sqrt{4-\cos^2 x}} \, dx$$ Using substitution $u^2 = 4 - \cos^2 x $ we get $$\int_\ \frac{2u }{\ u }…
davoid
  • 351
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Calculating an integral $\int_{0}^{1}x^k(1-x)^{n-k}dx$

I confused for calculating $$\int_{0}^{1}x^k(1-x)^{n-k}dx$$ one solution that I guess is: $$1^{n}=(x+1-x)^{n}=\binom{n}{k}x^{k}(1-x)^{n-k}$$ so $$x^{k}(1-x)^{n-k}=\frac{1}{\binom{n}{k}}$$ finally…
AMIR 1996
  • 75
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Is the Lebesgue integral the completion of integrals on step functions?

Lierre gave a very helpful insight at answer 5 on A "clean" approach to integrals. about what Riemann integrals are. My question relates to whether this can be extended to Lebesgue integrals. Lierre pointed out that Riemman integrals can be seen as…
Rory Allen
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Solving $\int\frac{1+\ln x}{x^2\ln^2 x}dx$ by method of parts

I need solve the following integral: $$\int\frac{1+\ln x}{x^2\ln^2 x}dx,$$ by the method of parts. My attempt: \begin{align*} \int\frac{1+\ln x}{x^2\ln^2 x}dx={}&\int\frac{dx}{x^2\ln^2 x}dx+\int\frac{\ln x}{x^2\ln^2 x}dx=\int\frac{dx}{x^2\ln^2…
Madrit Zhaku
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Evaluation of $ \int_{-\infty}^{\infty}\arctan (\frac 1{2x^2})\ \mathrm dx$

Evaluate $$\int_{-\infty}^{\infty}\arctan\left(\frac{1}{2x^2}\right)\mathrm dx$$ And how can I solve it using $$\sum^{\infty}_{x=-\infty}\arctan\left(\frac{1}{2x^2}\right)\quad\text{ and }\quad…
juantheron
  • 53,015
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Fast way to show that $\int_{0}^{1}x(1-x) \sin (n\pi x)dx = \frac{4}{(n\pi)^3}$, $n$ odd

I can compute and show that for odd values of $n$, $$\int_{0}^{1}x(1-x) \sin (n\pi x)dx = \frac{4}{(n\pi)^3}$$ fairly easily by expanding the quadratic, splitting the integrand and then evaluating each component using integration by parts. However,…
Trogdor
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How to evaluate the integral $\int_{-1}^1 e^{ax^2+bx+c\sqrt{1-x^2}}dx$

Can anyone show if the following integral can be evaluated in closed form? \begin{equation} \int_{-1}^1 e^{ax^2+bx+c\sqrt{1-x^2}}dx \end{equation} The variable $x$ can be replaced by $\cos{\theta}$, with corresponding change of the interval of…
khalatnikov
  • 815
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Convergence of Integrals of Exponential Functions

Let $f$ be a non-negative real valued function on $[a,b]$, and let $p:[a,b]\to(1,\infty)$ such that $f^p\in L^1([a,b])$. Let $p_n:[a,b]\to(1,\infty)$ be a (uniformly bounded) sequence of (step-)functions converging to $p$ wrt. the $L^1$-norm, i.e.…
sranthrop
  • 8,497
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Find area bounded by $(\frac{x}{3}+\frac{y}{6})^3-9xy=0$

$C = \{(x,y) \mid (\frac{x}{3}+\frac{y}{6})^3-9xy=0\}$. I want to find the area bounded by $C$ in first quadrant. Can you tell me how to solve exercises like this? I have no idea what integral I need to solve, because my curve is implicitly defined.
qwenty
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Computing $ \iint_{[0,1]^2} \frac{-x\ln(xy)}{1-xy} \mathrm dx \mathrm dy $

I would like to compute $$ \iint_{[0,1]^2} \frac{-x\ln(xy)}{1-xy} \mathrm dx \mathrm dy $$ Without going into detail, here is what I found: $$ \int_{0}^{1}(\int_{0}^{1} \frac{-x\ln(xy)}{1-xy} \mathrm dx ) \mathrm dy=\int_{0}^{1}(-\sum_{n=0}^{\infty}…
Chon
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Integrate $\int \frac{x^5 dx}{\sqrt{1+x^3}}$

I took $1+x^3$ as $t^2$ . I also split $x^5$ as $x^2 .x^3$ . Then I subsituted the differentiated value in in $x^2$ . I put $x^3$ as $1- t^2$ . I am getting the last step as $2/9[\sqrt{1+x^3}x^3 ]$ but this is the wrong answer , i should get…
Seeker1201
  • 359
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Is there a general approach to solve integrals of the form $\int\frac f{f'}$?

It is easy to solve integrals of the form $\int\frac{f'}f$ using the defintion of the natural logarithm: $\int \frac{f'(x)}{f(x)}\;\mathrm dx = \ln f(x).\ $ Is there a similar identity for the case $\int\frac f{f'}$?
FUZxxl
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Proving $\int_{0}^{1}\sqrt{\frac{1-x}{1+x}}dx=\frac{π}{2}-1$

Proving $$\int_0^1 \sqrt{\frac{1-x}{1+x}} \, dx= \frac{π}{2}-1$$ My attempt is: I assumed the $1-x=u$ $du =-dx$ $$\int_0^1 \sqrt{\frac{u}{2+u}}\,(-du)$$
E.H.E
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When Are We Allowed to Break Up A Triple Integral?

I was looking over the triple integral below: And I was wondering, when exactly are we allowed to break up a triple integral into the product of its components?
Robert
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