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 of $1/(1+x \tan(x))^2$

How would you solve the following integral? $$\int \frac{1}{(1+x\tan(x))^2} dx$$ Any help would be appreciated.
user34304
  • 2,749
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Changing order of integration limits

$$\int_{1}^{3} \int_{0}^y x+y-1 \, dx \, dy = 9$$ How would I change the order of integration here? Wouldn't this require two integrals? $$\int_{0}^{1} \int_{1}^3 x+y-1 \, dy \, dx + \int_{1}^{3} \int_{x}^3 x+y-1 \, dy \, dx = 9$$ Why does this…
Quaxton Hale
  • 1,258
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How find this integral $I=\int_{0}^{\frac{\pi}{2}}(\ln{(1+\tan^4{x})})^2\frac{2\cos^2{x}}{2-(\sin{(2x)})^2}dx$

Find the value: $$I=\int_{0}^{\frac{\pi}{2}}(\ln{(1+\tan^4{x})})^2\dfrac{2\cos^2{x}}{2-(\sin{(2x)})^2}dx$$ I use computer have this …
math110
  • 93,304
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How to find this integral $\int_{0}^{1}\ln\ln\bigl(1/x+\sqrt{(1/x^2)-1}\,\bigr)dx$

How do I compute this integral ? $$I=\int_{0}^{1}\ln{\left(\ln{\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)}\right)}dx$$ In the math chatroom someone suggests setting $x=\operatorname{sech}(t)$ and that the result immediately follows. I don't…
math110
  • 93,304
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Evaluate the integral $\int_{-2}^{2} \frac{1+x^2}{1+2^x}dx$

My friend asked me ot evaluate the integral: $$\int_{-2}^{2} \frac{1+x^2}{1+2^x}dx$$ And he gave me the hint: substitute $u = -x$. And so I did that, but I can't seem to get any farther than that. Could someone please provide some hints and help as…
Vishwa Iyer
  • 1,732
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Evaluate $\int \frac {x^2}{\sqrt{\arctan x}} dx$

Is there any closed form expression of $$\int \dfrac {x^2}{\sqrt{\arctan x}} dx?$$
Souvik Dey
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Prove the convergence of integral

$$\int^{\pi /2}_{0} \frac{\ln(\sin x)}{\sqrt x}dx$$ Use the segment integral formula? The $\sqrt x$ is zero at $x=0$ and $\ln\sin x$ is $-\infty$
zhen
  • 101
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Formalism in integration

Let's say we have some $y(t)$. The derivative of $y$ along time axis will be $y'(t)=\frac{dy(t)}{dt}=\frac{dy}{dt}$. So I will integrate like this over time: $\require{cancel}$ $\int_{t=0}^{+\infty}\frac{dy}{\cancel{d\tau}}\cancel{d\tau} =…
FELIPE_RIBAS
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slightly tricky integral

was asked to evaluate $\displaystyle\int_0^\infty \dfrac{\log(x)}{1+x^2} dx = I$ firstly, I got the solution using the substitution $ t = \dfrac{1}{x} $ and then getting $\displaystyle\int_0^\infty \dfrac{\log(x)}{1+x^2} dx =…
Warz
  • 1,129
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Calculus integration problem. [HW help]

Can someone please help me solve the following calculus problem for my homework?
1279343
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On integrating a "gaussian-like" integral

Let the following "gaussian-like" integral: $$ I = \int_{\Re^n} \! \frac{1}{(2\pi)^{n/2}|\Sigma|^{1/2}} \exp \left\{ -\frac{1}{2} (\mathbf{x}-\mathbf{\mu})^T\Sigma^{-1}(\mathbf{x}-\mathbf{\mu}) \right\} \mathbf{x} \,\mathbf{d}\mathbf{x}, $$ where…
nullgeppetto
  • 3,006
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If a function is integrable, then it is bounded

Probably a simple question, but I wonder about the following. I know that if a function $f : \mathbb{R} \rightarrow \mathbb{R} $ is (Riemann)integrable, then it is bounded. I wonder if I can generalize this to functions on $\mathbb{R}^3$ (now for an…
Rayman
  • 227
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Dirac Delta Constraint Question

Given an integral of the form \begin{equation}\int d\bar z\, dz\, \delta (\bar z \cdot A \cdot z-b)\,f(\bar z,z)\end{equation} Where $z$ is a complex $n$ dimensional vector, and $A$ is an arbitrary, possibly complex matrix, how would you sort out…
TeeJay
  • 1,303
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About an integral from MIT Integration Bee 2024

Good evening, I was interested in the third integral from the finals of the MIT Integration Bee 2024 : $$I = \int_{-\infty}^{\infty} \frac{1}{x^4+x^3+x^2+x+1} \hspace{0.1cm} \mathrm{d}x$$ One way to solve this is to identify that the denominator is…
LexLarn
  • 673
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Formula for integral of $x^a \ln^n(x) dx$

Can you explain this formula, I don't get how it works. Given $a > 0$, the integral is $$ \int_{0}^{1} x^a \, \ln^n(x) \, dx = \frac{(-1)^nn!}{(a+1)^{n+1}}. $$ I started integrating it (by parts) and understood that the first part is always $0$. So…
Allegrina
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