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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for every continuous function prove that $\int_0^{x} (t^3)f(t^2)dt = 0.5 \int_0^{x^2} tf(t)dt$

for every continuous function prove that $\int_0^{x} (t^3)f(t^2)dt = 0.5 \int_0^{x^2} tf(t)dt$ I thought about the Fundamental theorem of calculus but dont really know how to advance from there.
dani
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Integral $\int_0^\infty \frac{x \ln(1+x^2)}{e^{2 \pi x}+1}\,dx=\frac{19}{24}-\frac{23}{24}\ln 2-\frac{\ln A}{2}$

Edit: A friend of mine used a contour integral to solve this problem and his final answer matches with mine. It seems that Wolphram´s is incorrect I saw the following integral here and went to proof it. $$\int_0^\infty \frac{x \ln(1+x^2)}{e^{2 \pi…
Ricardo770
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Integrate $\int\limits_1^3 {\frac{{\left[ {{x^2}} \right]}}{{{{\left[ {{x^2} - 8x + 16} \right]}} + \left[ {{x^2}} \right]}}dx} = $

Solve $\int\limits_1^3 {\frac{{\left[ {{x^2}} \right]}}{{{{\left[ {{x^2} - 8x + 16} \right]}} + \left[ {{x^2}} \right]}}dx} = \_\_\_\_\_$ where [.] represent greatest integer function. My approch is as follow $\int\limits_1^3 {\frac{{\left[ {{x^2}}…
Samar Imam Zaidi
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Double integral - finding the region

I need to find the volume of an object bounded by the following planes: $$z=y+4$$ $$x^{2}+y^{2}=4$$ $$z=0$$ Is this true that the region of interest on the XY plane is the whole circle drew by the second formula above?
khernik
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Indefinite Integration of product of terms (of a series)

In this question product of terms was given so logarithm might help but it does not works.I used the partial fraction but there will be 2021 distinct fractions to be integrated so how to proceed further…
PHYSION
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Volume integral - finding the region

I need to find the volume of an object set with the following function: $$x + y + z = 1$$ And all three axis. So I converted the function into $z = -x - y + 1$, and it gave me kind of a clepsydra, crossing x,y plane in x = y = 1. So the region seems…
khernik
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Why is it wrong to prove the first fundamental theorem of calculus by geometric argument?

I read from the Apostol book calculus Vol 1 that the First fundamental theorem of calculus is as follows: Let $f$ be a function that is integrable on $[a,x]$ for each $x$ in $[a,b]$. let $c$ be such that $a\le c\le b$ and define a new function $A$…
Dachuan Huang
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Evaluate $\int \cfrac{150u^3}{e^{\pi u}-1}du$

When I was watching Vtuber content, I found this guy has an integral formula in his header of twitter. $$\int \cfrac{150u^3}{e^{\pi u}-1}du$$ It seemed there is no attempt to Evaluate this integral on the Internet, so I wanted to try it myself. My…
R.A
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Justify convergence of an integral

How would I justify the convergence of the following integral? $$\int_0^1 \frac{1}{1-x} + \frac{1}{\log(x)} dx$$ So far I looked at the laurent series of $1/\log(x)$ and I tried graphing the functions involved…
user581023
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Integration By Substitution： Why are the two results different?

The sphere: $x^2+y^2+z^2\leq a^2$ is intercepted by the cylindrical surface $x^2+y^2=ax$. Calculate the intercepted volume. Consider the intercepted volume of the upper hemisphere, and then multiply it by 2: $$D=\{(x,y):x^2+y^2\leq ax\}$$ Now…
Tim
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What happens to this theorem when we enlarge (or shorten) the set of elementary functions?

I'm having lectures on complex analysis, there is this theorem: I got a bit curious about the following: Having an antiderivative depends also on our choice of elementary functions, I am aware that we can enlarge (or shorten) this set (at least…
Red Banana
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Getting different answers when using integration by parts vs adding zero

$$\int \:bx\left(x+a\right)^{n-1}dx$$ I tried using $(x+a-a)$ which gives the apparently correct $$\frac{b}{n+1}\left(x+a\right)^{n+1}-\frac{ba}{n}\left(x+a\right)^n+C$$ However, I tried using by parts as well, using the tabular/DI method. It needed…
user71207
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Evaluate the integral of $\int \frac{1}{\left(\ln x\right)^{\ln x}}dx$

I recently completed a quiz for my Calc 2 class and I could not find the answer to this integral: $$ \int \frac{1}{\left(\ln x\right)^{\ln x}}dx $$ I tried to use integration by parts and was able to get to: $$ \int \frac{1}{\left(\ln x\right)^{\ln…
BlastKast
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How do I solve for $A$ such that $\int_{-\infty }^{\infty } A e^{-\lambda(x-a)^2}\, dx = 1$?

How do I solve for $A$ such that $\int_{-\infty }^{\infty } A e^{-\lambda(x-a)^2}\, dx = 1$? I first attempt u-substitution like so: Let $u = x-a$ $\implies du = dx$ and $x = u + a$, so $$ \int_{-\infty }^{\infty } A e^{-\lambda(x-a)^2}\, dx = …
user1068636
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$ \int_{- \frac{\pi}{4}}^{\frac{\pi}{4}} \tan^{-1}(e^{\tan(x)}) dx $ any hints?

I am struggling with following integral. $$ \int_{- \frac{\pi}{4}}^{\frac{\pi}{4}} \tan^{-1}(e^{\tan(x)}) dx $$ A small hint is enough.
easymathematics
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