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.

73636 questions
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Multiple integral: how to retrieve abscissa range

We have the double integral: $$\int \int_D 2x + 3y \; dx\;dy$$ The domain in which we want to calculate this is the flat region defined by the curves: $$y = x^2 \; ; \; y=x$$ Then, through the decomposition rules we resolve the internal integral to…
user3204810
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Integrate: $\int x(\arctan x)^{2}dx$

I'm not sure how to start I think we have to use integration by parts $$\int x(\arctan x)^{2}dx$$
Pie Man
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How to calculate $\int\frac{x}{x^2-x+1}\, dx$?

$$\int \frac{x}{x^2-x+1}\, dx = \int \frac{x}{(x-\frac 1 2)^{2} + \frac 3 4}\, dx = \int \frac{x}{(x-\frac 1 2)^2 + (\frac {\sqrt{3}} {2})^2}$$ Substitute $u= \frac{2x-1}{\sqrt{3}}, du=\frac{2}{\sqrt{3}}dx$: $$\frac {\sqrt{3}} 2 \int…
beforepim
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Evaluating a seemingly simple integral

I'm trying to evaluate the following integral, which arised while attempting to find the sum of a series : $$\int_{0}^{1} \frac{\ln(x)}{x-1} \ln(1+\sqrt{x})\text{d}x$$ I've tried unsuccessfully some substitutions, integration by parts, feynman…
Harmonic Sun
  • 1,952
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Evaluate $\int_0^\infty \cos(bx)(x-\ln(e^x-1))dx $

I have been given the integral $$\int_0^\infty \cos(bx)(x-\ln(e^x-1))dx$$ from a friend. I found an answer in terms of the digamma function, but he told me that the answer is obtainable without imaginary numbers. I am completely dumbfounded on…
Tom Himler
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Compute $\lim_{n\to \infty }\frac{1}{n}\sum_{k=1}^n\left(1+\frac{k}{n^2}\right)^n$

I want to compute $$\lim_{n\to \infty }\frac{1}{n}\sum_{k=1}^n\left(1+\frac{k}{n^2}\right)^n.$$ I really tried several thing, but this $\frac{1}{n^2}$ annoy me very much. It looks like a Riemann sum, but I can't conclude without more information.
user380364
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$\int{\cos(x)\cosh(x)+\sin(x)\sinh(x)}dx$

How would I evaluate the integral: $$I=\int{(\cos(x)\cosh(x)+\sin(x)\sinh(x)})\,dx$$ My thought was to use: $$\cos(ix)=\cosh(x)$$ and $$\sin(ix)=i\sinh(x)$$ or expand all four trig functions into exponentials but this was very messy EDIT: If I split…
Henry Lee
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How to solve the integration

How to find $$ \frac{\int_0^{\pi}x^3\log(\sin x)\,dx}{\int_0^{\pi} x^2 \log(\sqrt{2} \sin x)\,dx} $$ I couldn’t resolve it by using integration by parts.
Alex
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Is there a more elegant way of computing $\int \frac{1}{\sin(x)}dx$ and $\int \frac{1}{\cos(x)}dx$?

Both integrals can be solved by substitution, and while I am comfortable with that, in both cases I find the method unbearably ugly, mostly because there are hundreds of overtly feasible substitutions (and the corresponding factor the denominator…
Meow
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Simple proof for a simple case of Hardy's inequality

I am trying to prove Hardy's inequality in the case of 2, so: $$\int_0^\infty \frac{1}{x^2} \big(\int_0^x f\big)^2 dx\leq 2\int_0^\infty f^2$$ Where f is continuous over $\mathbb{R}_+$ and such that the integral of $f^2$ converges. The way I'm…
John Do
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Integrating both sides of equation

I'm given that -> $$\frac{dy(x)}{dx}=x$$ I integrate both sides -> $$\int \frac{dy(x)}{dx}dx=\int x\,dx$$ Would it be correct if i canceled out the $dx$s and wrote -> $$\int dy=\int x\, dx$$ therefore $$y= \frac{x^2}{2}+C,$$ where $C\in \mathbb{R}$
EldarRahim
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Integral $ \int \frac{\operatorname d\!x}{\sin^3 x} $

I need to calculate the following integral for my homework, but I dont know how. If someone show me step by step solution I would really appreciate it. $$\int \frac {1}{\sin^3(x)} dx$$
zdarsky.peter
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Evaluate $\int_{0}^{\pi\over 4}{\ln(\tan(x))\over \cos^{2n}(x)}\mathrm dx$

$$\int_{0}^{\pi\over 4}{\ln(\tan(x))\over \cos^{2n}(x)}\mathrm dx=F(n)\tag1$$ $n\ge1$ $F(1)=-1$ $F(2)=-{10\over 9}$ $F(3)=-{284\over 225}$ How do we evaluate the closed form for $(1)$? $u=\tan(x)$ then $\cos^2{(x)}\mathrm du=\mathrm…
gymbvghjkgkjkhgfkl
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Proving two integrals are equal to each other

Question: How do you show that$$\int\limits_{0}^{\infty}\frac {e^{-ax}}{\sqrt{b+x}}\,\mathrm dx=\int\limits_{\sqrt{ab}}^{\infty}\frac 2{\sqrt{a}}e^{ab-t^2}\,\mathrm dt$$ This problem emerged when I was trying to prove an identity by working…
Crescendo
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Evaluation of an integral involving sign function

Let $w>0$, $\lbrace k_{ij} \rbrace_{i,j\in [1,n]}$ be numbers taking either $0$ or $1$ as value and $i$ the usual complex number. I am trying to prove the following identity. $$\int_{]-\infty,0]^n}dx_1\dots dx_n\,…
Alexandre Krajenbrink
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