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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Compute $\int_{0}^{1}\left[\frac{2}{x}\right]-2\left[\frac{1}{x}\right]dx$

The question is to find $$\int_{0}^{1}\left(\left[\dfrac{2}{x}\right]-2\left[\dfrac{1}{x}\right]\right)dx,$$ where $[x]$ is the largest integer no greater than $x$, such as $[2.1]=2, \;[2.7]=2,\; [-0.1]=-1.$ Is there any nice method to solve this…
math110
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When does it hold that $\int_{0}^{x} fg=\left(\int_{0}^{x} f\right)\left(\int_0^x g\right)$

I was wondering when it held that $$\int\limits_0^x fg=\left(\int\limits_0^xf\right)\left(\int\limits_0^xg\right)$$ Let $$P:= x \mapsto \int\limits_0^x fg$$ $$F:= x \mapsto \int\limits_0^x f$$ $$G:= x \mapsto \int\limits_0^x g$$ The equality…
xavierm02
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$\int \frac{\sin^3x}{\sin^3x + \cos^3x)}$?

Is it possible to evaluate the following integral:$$\int \frac{\sin^3x}{(\sin^3x + \cos^3x)} \, dx$$
Legendre
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how to calculate the integral of $\sin^2(x)/x^2$

Possible Duplicate: Proof for an integral involving sinc function How do I show that $\int_{-\infty}^\infty \frac{ \sin x \sin nx}{x^2} \ dx = \pi$? $\int_{-\infty}^{\infty}\sin^2(x)/x^2=\pi$ according to wolfram alpha. That is such a beautiful…
ftiaronsem
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Can all integration be thought of as projections?

For example, the integral of the function f(x) could be thought of the projection of f on the function g, where g is identically 1. Following this logic, can we think of the multiplication of f and g as the area between f and g, no matter how…
Fraïssé
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LIATE : How does it work?

When doing Integration By Parts, I know that using LIATE can be a useful guide most of the time. For those not familiar, LIATE is a guide to help you decide which term to differentiate and which term to integrate. L = Log, I = Inverse Trig, A =…
Trogdor
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which is bigger $I_{1}=\int_{0}^{\frac{\pi}{2}}\cos{(\sin{x})}dx,I_{2}=\int_{0}^{\frac{\pi}{2}}\sin{(\sin{x})}dx$

let $$I_{1}=\int_{0}^{\dfrac{\pi}{2}}\cos{(\sin{x})}dx,I_{2}=\int_{0}^{\dfrac{\pi}{2}}\sin{(\sin{x})}dx$$ $I_{1}$ and $I_{2}$ which is biger? I hope see more nice methods,Thank you solution 1: note…
math110
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How to change order of integration in a double integral?

How would you go about changing the order of integration in a function say ; $$\int_0^8\int_\sqrt[3]{y}^2 f(x,y)~dx~dy$$
user90426
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Evaluating $ \int^{\infty}_{-\infty}\sin\left({\pi}^{4}x^{2}+\frac{1}{x^2}\right) dx$

$$\int^{\infty}_{-\infty}\sin\left({\pi}^{4}x^{2}+\frac{1}{x^2}\right) dx$$ This is a problem from the Pi Mu Epsilon Journal, and I'm having great trouble answering it. I've tried some substitutions and any trick I could think of to find some…
und
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$\int f(x)\,\mathrm{d}x = \left(\int_0^x f(t) \, \mathrm{d}t\right) + C$

If $f(x)$ is a continuous function on $\mathbb{R}$ and I am asked to find $\int f(x) \, dx$, what is the problem with the following answer: $$\int f(x)\,\mathrm{d}x = \left(\int_0^x f(t) \, \mathrm{d}t\right) + C$$
user17488
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Prove that $\Gamma(p) \cdot \Gamma(1-p)=\frac{\pi}{\sin (p\pi)}$ for $p \in (0,\: 1)$

Prove that $$\Gamma(p)\cdot \Gamma(1-p)=\frac{\pi}{\sin (p\pi)},\: \forall p \in (0,\: 1),$$ where $$\Gamma (p)=\int_{0}^{\infty} x^{p-1} e^{-x}dx.$$ Here's what I tried: We have $$B(p, q)=\int_{0}^{1} x^{p-1} (1-x)^{q-1}dx=\frac{\Gamma(p)\cdot…
Iloveyou
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Definite integral containing log and cot functions

Consider the following integral $$c=\int_0^{\pi/2}\log(1-x\cot x)\, \mathrm{d}x\approx-3.35333726288947201778500718670823032.$$ I suspect it can be analytically computed because by expanding the $\log$ function,…
yarchik
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Integral from infinity to infinity

My physics professor today wrote on the blackboard: $$ \int_{\infty}^{\infty} f(x) dx = 0 $$ for every function $f$. And the proof he gave was: $$ \int_{\infty}^{\infty} f(x) dx = \int_{\infty}^{a} f(x) dx + \int_{a}^{\infty} f(x)dx = -…
Victor
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How prove this integral equation $\int_{a}^{b}\frac{1}{\sqrt{|f(x)|}}dx=\int_{c}^{d}\frac{1}{\sqrt{|f(x)|}}dx$

let $a>b>c>d$,and $$f(x)=(x-a)(x-b)(x-c)(x-d)$$ show that $$\int_{a}^{b}\dfrac{1}{\sqrt{|f(x)|}}dx=\int_{c}^{d}\dfrac{1}{\sqrt{|f(x)|}}dx$$ my try: maybe let $$u=x+( )$$ such when $x=a,b$ then $u=c,d$? if $$a+d=b+c$$ then we take $$y=a+d-x$$ then we…
user94270
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Is there a math theorem by which a contour integral is equal to a double integral?

I was reading Maxwell's relations and came across: $$\oint pdV=\oint TdS\Rightarrow \iint dpdV=\iint dTdS.$$ I know this is straightforward to see since they both represent the surface area, but I've never seen a math theorem on textbooks that…
Lluvio Liu
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