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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visualization of the method of steepest descent

I am trying to understand the method of steepest descent (complex integral). I looked in some complex analysis books and also on Wikipedia, but I still don't understand the methodology of approximating such integrals nor the name of this…
bill
  • 151
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Show that $\int_0^1 \frac{\ln^3(x)}{1+x} dx = \sum_{k=0}^{\infty}(-1)^{k+1} \frac6{(k+1)^4}$

Show that; $$ \int_0^1 \frac{\ln^3(x)}{1+x} dx = \sum_{k=0}^{\infty}(-1)^{k+1} \frac6{(k+1)^4}$$ I arrived to the fact that $$ \int_0^1 \frac{\ln^3(x)}{1+x} dx = \sum_{k=0}^{\infty}(-1)^k\int_0^1 x^k \ln^3(x)dx$$ But I am unable to continue…
user361013
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Evaluate $\int_\frac12 ^2 \frac1x\tan(x-\frac1x)dx $

$$\int\limits_\frac{1}{2} ^2 \frac{1}{x}\tan\left(x-\frac{1}{x}\right)\mathrm{d}x$$ I have tried substitution and by parts and it seems failed at all. Can anyone give me some hints?
YIPYIP
  • 67
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What is the integral of $\int\frac{(x-1)e^{1/x}dx}{x}$?

I have been trying to solve this integral $\int\frac{(x-1)e^{1/x}dx}{x}$ I used WolframAlpha to solve it but it doesn't show the process. The solution is $e^{1/x}{x} + constant$
Miguel Alvarez
  • 105
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My conjecture $\int_{0}^{1}{x^n-1 \over \ln(x)}dx=\ln(n+1)$

$$\int_{0}^{1}{x^n-1 \over \ln(x)}dx=\ln(n+1)$$ Let deal with case $n=1$ $$I=\int_{0}^{1}{x-1 \over \ln(x)}dx=\ln(2)$$ $u=\ln(x)$ $\rightarrow du=\frac{1}{x}dx$ $x \rightarrow 1 ,u=0$ $x \rightarrow 0,…
gymbvghjkgkjkhgfkl
  • 8,090
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Integrate $\sqrt{x^2 + x}$

I'm stuck with trying to integrate the following expression: $$ \int (\sqrt{x^2 + x}) dx $$ I have tried u-substitution where $u = \sqrt{x}$, but it didn't get far. How should I approach this? Thanks ahead!
user321070
  • 193
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the integral :$ \int^\infty_{-\infty}e^{-x^{2}+x}dx $

$$ \int^\infty_{-\infty}e^{-x^{2}+x}dx $$ May be completing the square on the $-x^{2}+x$. Then, make a sub. then end up with the Gaussian integral with a constant multiple of $e^{\frac{1}{4}}$ ??
Frank
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Evaluation of $\int_{0}^{a} \frac{\sqrt{a+x}}{\sqrt{a-x}} dx$

Evaluate : $\int_{0}^{a} \frac{\sqrt{a+x}}{\sqrt{a-x}} dx$ My approach : I multiplied both sides by $\sqrt{a+x}$ and after simplification it comes down to : $\int_{0}^{a} \frac{a}{\sqrt{a^{2}-x^{2}}} dx + \int_{0}^{a} \frac{x}{\sqrt{a^{2}-x^{2}}}…
Mojo Jojo
  • 463
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Integrating, the $\int_0^\infty \frac{\text{ d}x}{x^2\ln x+1}$

I am trying to find a closed form for $$\int_0^\infty \frac{\text{ d}x}{x^2\ln x+1}$$ My usual tactic of partial differentiation under the integral (Mellin) does not work here due to that pesky +1
user323504
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Absolute value in integrating factor of First-Order Linear Differential Equation

Question states: $$ y' + \frac{y}{x} = 6x+2$$ Obviously x cannot be zero. If we assume that $x$ is positive (i.e. $x>0$), we find the integrating factor as $$u(x)=e^{\int \frac{1}{x} dx}$$ which is equal to $x$. Then the solution is $$y(x)=…
user304152
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Understanding of swapping bounds of integral

Please help me understand swapping the bounds of an integral better. I learned that $$\int_{a}^{b} f(x) dx = -\int_{b}^{a} f(x) dx$$ Now when I try to visualize this, take $\sin(x)$ for example, $\int_{\pi}^{2\pi} \sin(x) dx$ and $-…
user321070
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Evaluate the Integral.

Can someone help evaluate the following integral? (I have tried doing it with Mathematica but I just get the integral back - I am assuming that it can't figure out how to solve it?) \begin{equation} I=\int \frac{\csc ^2(\pi f) (\sin (2 \pi …
Saqib Shah
  • 113
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How can I find the area of the shadow?

Consider a lit candle placed on a cylinder. If the candle is placed at the center of the top surface, let the distance from the origin (center of the surface) to the end of the shadow be $r$. In this case the area of the shadow can easily be…
Green Noob
  • 875
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Find $\int\frac{dx}{\cos^3x-\sin^3x}$

$\int\frac{dx}{\cos^3x-\sin^3x}$ Let $I=\int\frac{dx}{\cos^3x-\sin^3x}=\int\frac{dx}{(\cos x-\sin x)(\cos^2 x+\sin^2 x+\sin x\cos x)}$ But it does not seem to be solved further by this method,so i tried another…
Brahmagupta
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Evaluate $\int_1^\infty \cosh^{-1}(x) \ln(x^2-1) \exp \left(- \frac{x}{T} \right) dx $

I would be interested in any clue on how to evaluate the following integral $$\int_1^\infty \cosh^{-1}(x) \ln(x^2-1) \exp \left(- \frac{x}{T} \right) dx $$ I have tried integration by parts but it seems to lead only to other integrals of the same…
chris
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