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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

A definite integral is defined as the area under a function from $a$ to $b$. Definite integrals sometimes involve calculating the indefinite integral, which is a function giving the area from $0$ to any $x$. However, definite integrals are most often separate from indefinite integrals in that the indefinite integral may not exist on its own. This is usually in the case of piece wise functions that are split along certain key points or integrals involving asymptotes.

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Why is it obvious that this integral is undefined?

If we have the following integral: $$\int_0^1\frac{1}{\left(x+a\right)\left(x+2\right)}\,dx$$ Why is it that it is obvious that this integral will be undefined for $0>a>-1$? I have a rough idea that one knows that this creates an asymptote at…
Benjamin
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Integrating both sides of stochastic equation

When I was reading one document, I noticed one integration there, which I found very odd. http://www-2.rotman.utoronto.ca/~hull/TechnicalNotes/TechnicalNote22.pdf I am referring to the part: "Integrating between time 0 and time T". If I understand…
siwy9
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Infinite Dimensional integral

Is an integral over an infinite dimensional space defined somewhere? For example, does it make sense to think about $$\lim_{n \to \infty} \int_{\mathbb{R}^n}f_n(\mathbf{x})\,d\mathbf{x}, \quad \mathbf{x} \in \mathbb{R}^n, \quad f_n:\mathbb{R}^n \to…
Student
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Questions About Integrals

I've started moving into integral calculus and I am a bit confused about double and triple integrals. If double integrals are for finding the area of a 3 dimensional space between a graph what do triple integrals do? Sorry if this question doesn't…
NooneLikesDoubleIntegrals
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How to calculate these integral

Find the value of $$\int_{0}^{x} \lfloor \cot(t)\rfloor\,dt$$ where $x \in [(4n+1)\frac {\pi}{2},(4n+3)\frac {\pi}{2}]$ and $n\in N $ Also $\lfloor\cdot \rfloor$ represnts the greatest integer function. I tried this by graphical method but i am…
Girish Kumar Chandora
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Proving $ \int_0^\infty \frac { e^{-ax} - e^{-bx} } x dx = \ln \frac b a $

How can we prove this identity? (Taken from Wikipedia's List of Definite Integrals) $$ \int_0^\infty \frac { e^{-ax} - e^{-bx} } x dx = \ln \frac b a $$ It seems that the integral is not elementary and $$ \int_0^\infty \frac { e^{-ax} - 1 } x dx…
Adrian Savage
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Is it possible for integrals to have functions of the integrand in the bounds of the integral such as $\int_0^x xdx$?

Is it possible for integrals to have functions of the integrand in the bounds of the integral such as $\int_0^x x\,dx$ or more generally $\int_0^{f(x)} x\,dx$ ? I have seen this several times in the past few weeks for the first time and I am not…
IntegrateThis
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How to show the inequality for $\int_{0}^{\pi/4}\tan^n(x) dx$

I can't seem to figure out the inequality option, I substituted n=2 for the answer bit is there any elegant way of getting it?
TheLostGuardian0
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Why does $\ln(x)\ln(1-x)\mid^1_0=0$

$$\ln(x)\ln(1-x)\mid^1_0=0$$ I see this result once or twice a day in various forms without proof. Could someone just write the proof so I don't embarrass myself if someone on the street or in a grocery store asks me? One comment for the proof was…
user5389726598465
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Area between $y=x^{6} - x$ and $x = y^{6} - y$?

I realize that they are the same equations but with $x$ and $y$ switched, but I'm stuck from there on how to approach the problem.
user109987
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Let $ f(x) $ be the solution of $ \ \int_{0}^{x}e^{(x-t)} f(t) dt=x, \ \ x>0 $.

Let $ f(x) $ be the solution of $ \ \int_{0}^{x}e^{(x-t)} f(t) dt=x, \ \ x>0 $. Then $ f(1)= ? $ (a) $-1 $, (b) $ 0 $, (c) $ 1 $, (d) $ 2 $, Ans. $ \int_{0}^{x}e^{(x-t)}f(t)dt=x $. Differentiating under integral sign , we get Any help ?
MAS
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what substitution do i use to solve this triple integral?

I have a density function $p(x, y)=(x^4 +y^4)(x^2 +y^2)$ and i need to calculate mass of tile that lies in first quadrant between $1< xy<3$ and $1< x^2 - y^2<3$ . I think a change of variable to polar would help me, but cant figure how to make…
naftoli
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Difficult to start a triple integral

I have to calculate the integrale of $f$ on $T$. I think it is a triple integrale, but I didn't manage to solve it. $$ f(x,y,z)=\frac{1}{1+x+y+z} $$ $$ T=\{(x, y, z)\in{(\mathbb R^+)^3}:||(x, y, z)||_1\le{1}\} $$ With $$ T=\{(x, y, z)\in{(\mathbb…
Geoffrey Gauthier
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Definite Integral between bounds $a$ and $b$ = 0?

if $g(2) = 4$ and $g(3) = 5$, then $\int_2^3 (g'(x) - 1) dx$. i have solve this to be 0 since the anti-derivative of g'(x) - 1 is g(x) - x, therefore, $[g(x) - x]_2^3$ = 0. but how can the area between x = 2 and x = 3 for (g'(x) -1) be 0?
lohboys
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Logarithmic integral--

$$\int_c^d\int_a^b\ln|x-y|\ dx\ dy$$ Here we have double integral of a logarithmic function. I really worked hard but nothing worked out. Any kind of insight, approximation or even document suggestion towards solution of this integral equation will…
ömer
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