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
13
votes
2 answers

How to prove that $\int_0^{\infty} \log^2(x) e^{-kx}dx = \dfrac{\pi^2}{6k} + \dfrac{(\gamma+ \ln(k))^2}{k}$?

I was answering this question: $\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$ and in my answer, I encountered the integral $$\int_0^{\infty} \log^2(x) e^{-kx}dx$$ which according to WolframAlpha for $k=1,2,3$ and thereby generalizing gives…
user17762
13
votes
3 answers

Polygon on a grid

Given a square constructed on a grid of points with integer coordinates, what is its maximum area, if we know that there are exactly 3 grid points in its interior? I have no idea how to start. I googled and found there is a Pick's theorem, but…
Wang Xiu Ying Zhang
  • 329
13
votes
1 answer

How to integrate $\int_0^{\pi/2}\frac{\sin^nx}{\sin^nx+\cos^nx}dx$?

Possible Duplicate: How can I calculate $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt$? How can we integrate $$\int_0^\frac{\pi}2\frac{\sin^nx}{\sin^nx+\cos^nx}\,\mathrm dx , \,\,\,\,\,\,\,\,\, n\in N \quad?$$ Thanks for any hint.
M.H
  • 11,498
  • 3
  • 30
  • 66
13
votes
3 answers

Closed-form for rational log integral: $\int_0^1\left(\frac{\ln x}{1-x}\right)^{n}dx$

If I may, I have a rather challenging integral. I am not so sure there is a closed form. $$\int_0^1 \left(\frac{\ln x}{1-x}\right)^n \; dx$$ I have evaluated when $n=1$ and $2$. But, when $n=3, 4, 5,\ldots$, the solution involves varying zeta…
Cody
  • 14,054
12
votes
2 answers

Gaussian integrals over a half-space

Edit: I shall try to reformulate my question in order to make it -hopefully- more clear. Let $X$ be a random variable that follows the $n$-dimensional Gaussian distribution. The probability density of $X$ is given by the following…
nullgeppetto
  • 3,006
12
votes
4 answers

How to find the exact value of the integral $ \int_{0}^{\infty} \frac{d x}{\left(x^{3}+\frac{1}{x^{3}}\right)^{2}}$?

$\textrm{I first reduce the power two to one by Integration by Parts.}$ $\begin{aligned}\displaystyle \int_{0}^{\infty} \frac{1}{\left(x^{3}+\frac{1}{x^{3}}\right)^{2}} d x &=\int_{0}^{\infty} \frac{x^{6}}{\left(x^{6}+1\right)^{2}} d…
Lai
  • 20,421
12
votes
1 answer

Evaluate $\int_{0}^{\frac{\pi}{2}} \ln(1+\sin^3 x)\text{d}x$

Here's the integral that I would like to solve. Purely for recreational purposes: $$I=\int_{0}^{\frac{\pi}{2}} \ln(1+\sin^3x)\text{d}x$$ Here's my shot at it. I would like to stick to this method if possible. Let $I(\alpha)$ be defined as…
Moni145
  • 2,142
12
votes
3 answers

Failing a basic integration exercise; where did I go wrong?

(This is a basic calculus exercise gone wrong where I need some feedback to get forward.) I've attempted to calculate an integral by first integrating it by parts and then by substituting. The result I got is not correct though. Can I get a hint…
Raudus
  • 223
12
votes
3 answers

How to solve this integral/better way to approach?

$$\int_{0}^{c} dy \sqrt{\frac{c-1/2y^2+1/3y^3}{1+2y}}$$ where c is a constant. This is coming from trying to find the area $$\int_{U \le c} dq_1dq_2$$ where $$U=\frac{1}{2}(q_1^2+q_2^2)-\frac{1}{3}q_2^3+q_1^2q_2$$ bounded by energy $c=U(q_1,q_2)$.
Некто
  • 189
12
votes
3 answers

Showing $8.9<\int_3^5 \sqrt{4+x^2} \, \mathrm d x < 9$

I am asked to show that $$8.9<\int_3^5 \sqrt{4+x^2} \, \mathrm d x < 9$$ I tried computing the integral but I end up with $$\frac 5 2 \sqrt{29} - \frac 3 2 \sqrt{13} + 2 \log{\left(\frac {5 + \sqrt{29}} {3+\sqrt{13}}\right)}$$ which isn't really…
user85798
  • 1
12
votes
4 answers

Constant of integration change

So, sometimes the constant of integration changes, and it confuses me a bit when and why it does. So for example, we have a simple antiderivative such as $$\int \frac{1}{x} dx $$ and we know that the result is $$\log|x| + C$$ and the domain is…
chilliefiber
  • 407
12
votes
3 answers

Is there a closed form for $\int x^n e^{cx}\,\mathrm dx$?

Wikipedia gives this evaluation: $$ \int x^ne^{cx}\,\mathrm dx=\frac1cx^ne^{cx}-\frac nc\int x^{n-1}e^{cx}\,\mathrm dx=\left(\frac{\partial}{\partial c}\right)^n\frac{e^{cx}}{c}$$ But I have no idea how I should exactly understand the partial part:…
johanvdw
  • 243
12
votes
3 answers

Definite Integral and Constant of Integration

I understand that when we are doing indefinite integrals on the real line, we have $\int f(x) dx = g(x) + C$, where $C$ is some constant of integration. If I do an integral from $\int f(x) dx$ on $[0,x]$, then is this considered a definite…
Ravi Thiagarajan
  • 121
11
votes
3 answers

show that $\int_0^1 (\sqrt[3]{1-x^7}-\sqrt[7]{1-x^3}) \, dx=0$

show that $$I=\int_0^1 (\sqrt[3]{1-x^7}-\sqrt[7]{1-x^3}) \, dx=0$$ I find this is Nice equalition! My try: let $$\sqrt[3]{1-x^7}=t\Longrightarrow x=\sqrt[7]{1-t^3}$$ so $$dx=-\dfrac{3}{7}t^2(1-t^3)^{-\dfrac{6}{7}} \, dt$$ so $$I=\frac{3}{7}\int_0^1…
math110
  • 93,304
11
votes
1 answer

Boku No Hero ep 80 Integral

I just managed to catch up to an anime called Boku No hero, and to my surprise they showed an integral in ep 80-81 I think. Where the main character tried to evaluate it and got $\frac{107}{12}$ but the answer was $\frac{107}{28}$. I'm just curious…
user8700908
  • 413
1 2 3
…
99 100