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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Simplifying an integral by changing the order of integration

Question: Consider a triple integral of the following form \begin{equation} \int_{x=0}^1 \int_{y=0}^{1} \int_{z=0}^{1} f(x,y,z)dzdydx. \end{equation} Because of the specific $f(\cdot,\cdot,\cdot)$ function I am dealing with, I would like to convert…
emper
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Using the substitution method for a simple integral

I have been playing with the substitution rule in order to test some ideas with computational graphs. One of the things I'm doing is applying the substitution to well known, and easy, integrals. For example, let's use that method to find the…
echo66
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Mechanical Surface Integrator

In an episode of "Dirty Jobs" Mike Rowe visited a tannery where they used an old mechanical device to calculate the surface area. Video shown [here] How is it calculating the surface area? Is it doing Riemann sum as it passes through with the width…
sheppa28
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How is it justified in indefinite integral to make trig substitutions whose range is not all real numbers?

How in indefinite integral we can do the substitution without bothering about the variable or function we are substituting the variable with? For example, it is very common to make trigonometric substitutions for evaluating indefinite integrals like…
Matt
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Evaluating $\int \frac{1-7\cos^2x}{\sin^7x\cos^2x}dx$

How do i evaluate $$\int \frac{1-7\cos^2x}{\sin^7x\cos^2x}dx$$. I tried using integration by parts and here is my approach $\int \frac{ sinx}{(1-cos^2x)^4\cos^2x} dx$ and then put $cos x=t$ and then tried to use partial fractions.I applied similar…
Navin
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Finding volume using integration

Find the volume generated when the region bounded by $y=x^3$ and the $x$-axis between $x=2$ and $x=7$ is rotated through $360^{\circ}$ about the $x$-axis. Here is my attempt is this correct? \begin{align*} y &= x^3 \, , \; 2\le x\le7 \\ V &=…
Dan
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In what situations is the integral equal to infinity?

In the following integral, p(x) and q(x) are probability distributions. Can you help me to determine in what situation this integral is equal to infinity. For example, I think that such a situations is when only p(x) has an infinite…
Marco
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Calculate $\int_{0}^{\infty }e^{-x^{2}-x^{-2}}dx$

How to calculate $$\int_{0}^{\infty }e^{-x^{2}-x^{-2}}dx$$ I have no idea where to start.Is it connect with the euler-poisson integral?
user308493
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Integral of sinc function multiplied by Gaussian

I am wondering whether the following integral $$\int_{-\infty}^{\infty} \frac{\exp( - a x^2 ) \sin( bx )}{x} \,\mathrm{d}x$$ exists in closed form. I would like to use it for numerical calculation and find an efficient way to evaluate it. If…
jaian
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Using an integration technique:$ \int^b_a \left[f(x)+f^{-1}(x)\right]dx=b^2-a^2$

I read about this integration technique on quora: If $a,b$ are fixed points of $f$, then $$ \int^b_a \left[f(x)+f^{-1}(x)\right]dx=b^2-a^2$$ Apparently it was used in the final of the 2013 MIT Integration bee but I can't find that question…
bigfocalchord
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Integrate $\int_0^{\pi/2} {\sin x \cos x \sqrt{\tan x} \ln{\tan x} \,dx}$

Challenge question: solve $$\int_0^{\pi/2} {\sin x \cos x \sqrt{\tan x} \ln{\tan x} \,dx}$$ It's a generalization of a recent Math.SE question, but how would one normally approach it?
THA
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Alternative ways to prove that $\int_0^{2\pi} \cos(\sin{x})e^{\cos{x}} dx=2\pi$ without complex analysis?

$$\int_0^{2\pi} \cos(\sin{x})e^{\cos{x}} dx=2\pi$$ I derived this rather incredible result via Cauchy's theorem as I was working through some simple contour integrals. I was wondering if this integral can be solved without complex analysis, and if…
user1892304
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Calculate $\int_0^\infty\frac{\sin(x)\log(x)}{x}\mathrm dx$.

Calculate $\displaystyle\int_0^\infty\dfrac{\sin(x)\log(x)}{x}\mathrm dx$. I tried to expand $\sin(x)$ at zero, or use SI(SinIntegral) function, but it did not work. Besides, I searched the question on math.stackexchange, nothing…
Faye Tao
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The integral $\int\frac{2(2y^2+1)}{(y^2+1)^{0.5}} dy$

What is $$\int\frac{2(2y^2+1)}{(y^2+1)^{0.5}} dy?$$ I split it as $\frac{y^{2}}{(y^2+1)^{0.5}} + \sqrt{y^2+1}.$ Now I substituted $y^{2}=u $ thus $2y\,dy=du$ so we get $0.5 \sqrt{\frac{u}{u + 1}} + 0.5 \sqrt{\frac{1 + u}{u}}$ but now what to do?…
Archis Welankar
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Evaluate $\int \frac{\sin^4 x}{\sin^4 x +\cos^4 x}{dx}$

$$\int \frac{\sin^4 x}{\sin^4 x +\cos^4 x}{dx}$$ $$\sin^2 x =\frac{1}{2}{(1- \cos2x)}$$ $$\cos^2 x =\frac{1}{2}{(1+\cos2x)}$$ $$\int \frac{(1- \cos2x)^2}{2.(1+\cos^2 2x)}{dx}$$ $$\frac{1}{2} \int \left[1-\frac{2 \cos2x}{1+\cos^22x}\right] dx$$ What…
Aakash Kumar
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