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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minimization of integral of multi-variable function

Let $\alpha \in (0,1) $ be some parameter which we can choose and $Y$ be some random variable (e.g. standard normal distribution), then we can define a function as below (where $v,e < 0$ are two variables): $S(Y,v,e; \alpha) = -\frac{1}{\alpha…
Selos
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Result from J Edwards Calculus Book

In Edwards classic treatise, p200, the following result is asked to be proved. Prove that $$ \int_0^{a} \frac{a}{(x+\sqrt{a^2-x^2})^2}dx = \frac{1}{\sqrt{2}}\ln(1+\sqrt{2})\ $$ I have made the obvious substitution $x = a \sin\theta$ but fail to…
Callie12
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Proving that $\int_0^\infty\frac{x^2 {\rm d} x}{e^x+1} = \frac{3}{4}\int_0^\infty\frac{x^2 {\rm d}x}{e^x-1}$ without zeta functions.

I know that it is possible to show that $\int_0^\infty\frac{x^2 {\rm d} x}{e^x+1} = \frac{3}{2}\zeta(3)$ and $\int_0^\infty\frac{x^2 {\rm d} x}{e^x-1} = 2\zeta(3)$ by rewriting the integrals to an expression containing the definition of the zeta…
Ewoud
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What did I do wrong in u-substitution

The problem: find the indefinite integral of $x(x-1)^2$ I used u-substitution, $u = x-1, x = u+1, du = dx$. which gave me $(u+1)u^2$. I distributed and got $u^3 + u^2$, and took the integral to get $[(u^4)/4] + [(u^3)/3]$ replacing $u$ gave me an…
Daniel B.
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Can you integrate without a $dx$

Long ago I realized that manipulation of derivatives was possible using algebraic quantities. One could take a differential instead of derivatives $$ d[\sin x]=\cos x\ dx $$ $$ \frac{d[\sin x]}{dx}=\frac{\cos x\ dx}{dx}=\cos x $$ My question is…
eschavez
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Integral? $\int_1^{\infty} \frac{x-1}{x^s -1} \frac{\,dx}{x}$

In looking at the sum $g(s) = \sum_{n=2}^{\infty} \frac{\zeta(n)}{s^n}$ for $|s|>1$, I found that $g(s)$ may be expressed as, $$\int_1^{\infty} \frac{x-1}{x^s -1} \frac{\,dx}{x}$$ and this is the integral I am trying to solve. Wolfram alpha appears…
Ryan Goulden
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Computing an intergral on a sphere

I'm struggling with the following integral: $$\int \limits_{S_t(x)} y_1^2 + y_2^2 + y_3^2 \, dA(y),$$ where $S_t(x) = \{y \in \mathbb{R}^3: \lvert y - x\rvert = t \}$. I understand the integral above as average value of the integrand over $S_t(x)$.…
Hendrra
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Is there a general solution to the integral $\int \frac{f(x)}{f'(x)}dx$?

I am trying to find a general solution in terms of $x$ and $f(x)$ of the integral $$ \int \frac{f(x)}{f'(x)}dx$$ I tried partial integration, substitution and tried using the fact that $\frac{1}{f'(x)}$ is equal to $(f^{-1})'(y)$. If the general…
Samuel
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Is there an integrable function mapping a closed interval to an open interval

Is there a function $f$ that is integrable on a closed interval $[a,b]$ for $a, b \in \mathbb{R}$ and maps this interval to an open but bounded interval? Edit: I should have specified better. What I wanted was more of a function $f$ with a primitive…
kyticka
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What is $\int_0^{\infty} (-1)^{\lfloor x^2 \rfloor} \, \operatorname{d}\!x \ ?$

What is $$\int_0^{\infty} (-1)^{\lfloor x^2 \rfloor} \, \operatorname{d}\!x \ ?$$
ORBOT Inc.
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Bounding $I(x)=\int_{0}^{x}\frac{(x-t)^2\exp(t)}{2}\mathrm dt$

Suppose $\forall x \in \mathbb{R}$, $I(x)=\displaystyle\int_{0}^{x}{\dfrac{(x-t)^2\exp(t)}{2}\,\mathrm dt}$. Without calculating $I(x)$ how can I prove that: $\forall x \ge 0$, $\quad0\le I(x) \le \dfrac{\exp(x)x^3}{6}$; $\forall x\le0$, $\quad…
pourjour
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Is it possible to compute $\int_0^x\frac{t^2dt}{(t\sin(t)+\cos(t))^2}$?

One of my students gave this integrale. By parts makes it more complicate. I see no classical substitution. Here is the integral to find $$\int_0^x\frac{t^2dt}{(t\sin(t)+\cos(t))^2}$$ Thanks in advance.
hamam_Abdallah
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Why can't I u-sub for the second part of this integral $\int \ln(2x+4)$?

I was reviewing one of my integral problems, and I ran into this problem: $$\int \ln(2x+4)dx $$ I did integration by parts and got this: $$\ln(2x+4)x - \int \frac{2x}{2x+4}du $$ For the second integral, I did a u-sub: $$ \frac{1}{2} \int…
mohhamed rafi
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Proving $\int_0^\infty e^{-x^p} dx$ converges or not

How can we prove that the following integral: $$\int_0^{+\infty} e^{-x^p} dx$$ converges or not? ($p$ is any given number) Based on what I've done so far, I think we should separate the cases $p \ge 1$ and $0 < p < 1$ in order to solve the question…
Jigsaw
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Recursion Formulas of $\int x^{\alpha}\ln x \ \text{dx}$ and $\int\frac{\ln^{\beta}x}{x} \ \text{dx}$

Suppose that I have recursion formulas of $\int x^{\alpha}\ln x \ \text{dx}$ and $\int\frac{\ln^{\beta}x}{x} \ \text{dx}$, and suppose that i found them(integration by parts), $$\int x^{\alpha}\ln x \ \text{dx}=\frac{x^{{\alpha} + 1}}{{\alpha} + 1} …
Sandra West
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