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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If $f(x) = e^{-x}+2e^{-2x}+3e^{-3x}+\cdots$ then find $\int_{\ln2}^{\ln3}f(x)dx$

Solve the following If $f(x) = e^{-x}+2e^{-2x}+3e^{-3x}+\cdots $ Then find $\int_{\ln2}^{\ln3}f(x)dx$ I don't have any idea.
kalpeshmpopat
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Integral $\int_{-\infty}^{\infty} \frac{\cos t \sin( \sqrt{1+t^2})}{\sqrt{1+t^2}}dt$

I want to evaluate the integral $$\int_{-\infty}^{\infty} \frac{\cos t \sin( \sqrt{1+t^2})}{\sqrt{1+t^2}}dt$$ I tried using Feynman’s trick and introduced the parameter $e^{-\sqrt{1+t^2} x}$, then differentiating with respect to $x$ under the…
meiji163
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Evaluating $\int_0^{\infty } \frac{x^{\alpha}}{\left(A+x^3\right) \left(B+e^x\right)} \, \mathrm dx \quad \alpha = \frac{9}{2},3,\cdots $

I'm looking for an analytical solution of the following two integrals $$\int_0^{\infty } \frac{x^{9/2}}{\left(A+x^3\right) \left(B+e^x\right)} \, \mathrm dx$$ and $$\int_0^{\infty } \frac{x^3}{\left(A+x^3\right) \left(B+e^x\right)} \, \mathrm…
Rainer
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How did you prove this integral?

This is my problem, true or false? prove that: $$\int_{0}^{\infty}\frac{\displaystyle{\sum_{k=1}^{\infty}}k\sin(kx)\,e^{-tk^2}}{\displaystyle{\sum_{k=1}^{\infty}}\cos(kx)\,e^{-tk^2}}dt=\frac{\pi^2({\pi-x})}{8}$$ and $0
math110
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Integral $\int\frac{6^x}{9^x-4^x}dx $

How to solve this integral: $$\int\frac{6^x}{9^x-4^x}dx $$ (I notice that $\frac{6^x}{9^x-4^x}=\frac{2^x3^x}{(3^x-2^x)(3^x+2^x)}$) Thank you!
Sandra West
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Evaluating :$\int\cot^{-1} (x^2+x+1)dx $

How to evaluate :$$\int\cot^{-1} (x^2+x+1)dx $$
dia
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Integral of $\frac{\sqrt{x^2+1}}{x}$

So I have to do the integration of $\frac{\sqrt{x^2+1}}{x}$. Give me a hint. What should I replace? Should I do it with integration by parts?
gfg
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Integral representation for Euler constant

I am trying to show that $$\gamma=\int_0^{\infty}\frac{\ln(1+x)+e^{-x}-1}{x^2}\;dx$$ I haven't done so much, but since this is related to $\psi (1)=\Gamma'(1)$ I have tried to work backwards. So $$\Gamma(x)=\int_0^{\infty}…
user556151
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Integral $\int_{-2}^0 \frac{x}{\sqrt{e^x+(x+2)^2}}dx$

I am trying to evaluate $$\int_{-2}^0 \frac{x}{\sqrt{e^x+(x+2)^2}}dx$$ So far I had no succes using trig substitution or integration by parts, also some random substitution like $x=2t$ and moved the exponential to the numerator, but I am stuck.…
user556151
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How to integrate $|x| \cdot x$

How to integrate this manually? $$ \int |x|\cdot x ~dx $$ My tries so far: $$ \int |x|\cdot x ~dx = (x^2/2)\cdot|x| - \int (x²/2)\cdot \mathop{\mathrm{sign}}(x) ~dx $$ Trying it again, but using sign(x) as first parameter, because sign(x) is not…
little raspy
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What is the easiest way to integrate $\left(\frac{1-x}{1+x}\right)^{1/2}?$

This is an indefinite integral that's supposed to be very easy: $$I=\int\sqrt{\frac{1-x}{1+x}}\,dx$$ I can only think of one way of calculating it, and it's a bit complicated, that is: substitute $x=\sin u$, and obtain $dx=(\cos u)\,du$ and…
Bartek
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Computing the indefinite integral $\int x^n \sin x\,dx$

$\newcommand{\term}[3]{ \sum_{k=0}^{\lfloor #1/2 \rfloor} (-1)^{#2} x^{#3} \frac{n!}{(#3)!} }$ I am trying to prove that for $n \in\mathbb N$, $$ \int x^n \sin x \, dx = \cos x \term{n}{k+1}{n-2k} + \sin x \term{(n-1)}{k}{n-2k-1} $$ I started with…
Gunnar
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Example where $\int |f(x)| dx$ is infinite and $\int |f(x)|^2 dx$ is finite

I read in a book that the condition $\int |f(x)|^2 dx <\infty$ is less restrictive than $\int |f(x)| dx <\infty$. That means whenever $\int |f(x)| dx$ is finite, $\int |f(x)|^2 dx$ is also finite, right? My understanding is that $|f(x)|$ may have a…
JACKY88
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trouble with this integral

Could anyone help me to do this integral ? $$\int_{\,0}^\infty \; \frac{\exp \left( -\frac{1}{x} -x\right)}{\sqrt{x}} \, dx = \sqrt{\pi}e^{-2} $$ I think you start with completing the square in the exponent, but what substitution do you make then ?…
Joe King
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Why is $\int_{0}^{1}x^x(1-x)^{2-x}\sin(x\pi)dx=\int_{0}^{1}x^{1+x}(1-x)^{1-x}\sin(x\pi)dx?$

I don't know why these two integrals yield the same results. $$\int_{0}^{1}x^x(1-x)^{2-x}\sin(x\pi)dx=\int_{0}^{1}x^{1+x}(1-x)^{1-x}\sin(x\pi)dx$$ Any hints, clues and ideas how to go about dealing these integrals(showing that they are the same and…
user348832
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