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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Evaluating $\int_0^{\pi/2} \frac{a}{a^2+\cos^2 \theta} \, d\theta$

I want to evaluate $$ \int_0^{\pi/2} \frac{a}{a^2+\cos^2 \theta} \, d\theta $$ and here is what Wolfram alpha gave me: $$ \int_0^{\pi/2} \frac{a}{a^2+\cos^2 \theta} \, d\theta=\frac{\tan^{-1} \left(\frac{a \tan…
user160738
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Is there a definite integral that evaluates to the constant $e$?

The integrand should not involve the constant $e$ itself nor, preferably, $\cosh$, $\sinh$, etc. $\pi$ arises in definite integrals such as $$\int_0^a \frac{dx}{\sqrt{a^2-x^2}} = \frac{\pi}{2}$$ The integrand must be an algebraic function with…
Inquitor
  • 85
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What is the indefinite integral $\int \sqrt{1-2\sqrt{x-x^2}} \ \mathrm dx$?

What is $$\int \sqrt{1-2\sqrt{x-x^2}} \ \mathrm dx$$ I have tried substituting everything and it doesn't seem to be working. Substituting trigonometry doesn't seem to work either.
Bob Jane
  • 61
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What IS the value of a complex line integral?

The value of a line integral in vector calculus can be seen as "the sum of how much the of a vector valued function affects every point of the curve". I know this is not a mathematical definition, but it helps in giving a mental picture of what the…
Data
  • 61
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How do I calculate: $\int \frac{dx}{3\sin^2 x+5\cos^2x}?$

How can I calculate this integral ? $$\int \frac{dx}{3\sin^2 x+5\cos^2x}=\text{?}$$ Thank you! I've tried using universal substitution but the result was too complicated to be somehow integrated. Can you please give me a useful hint ?
wonderingdev
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Integrating $\int_0^{\pi/2}\log^2(\sin^2x)\sin^2x{\rm d}x$

after lots of discussion and help from @Chris'sis I have tried integrating it in varied ways but it involves something that is new to me: $$ I=\int_0^{\pi/2}\log^2(\sin^2x)\sin^2x{\rm d}x$$ It could be written as: $$I=\int_0^{\pi/2}4\log^2(\sin…
RE60K
  • 17,716
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Upper and lower bound on integral

Consider the following integral $$\int_0^1 (1-x^n)^M \,d x$$ It converges to $0$ as $M\to\infty$, but I would like to find bounds on the convergence rates. What I mean is that it is straightforward to find constants A and B such that…
Ivan
  • 2,324
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How to find this integral?

I am trying to find the integral $$\oint_c Re(z)\;dz$$ where c is a circle $$|z|=2$$ I don't know what to do. I tried some things but I don't know if I am correct. $e^{i\theta} = \cos \theta +i \sin \theta$, so $Re(z) = \cos \theta$? And then…
tripons
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Solve $ \int_{0}^{\infty} \sin^2 \left(\frac{1}{x}\right)\mathrm{d}x$

I think this integral does not converge. I want to estimate downward the integral, but don't know how to.
Scheider
  • 51
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Yet another multivariable integral over a simplex

Let $p$ be a positive integer, let $(q_0,q_1,\cdots,q_p)$ be a sequence of positive integers and let $\beta \neq 1/2$ be a positive number. Then let $B>A>0$. The question is to prove the following identity for a integration over a $p$ dimensional …
Przemo
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Which integrals can be solved using Feynman's Technique?

How to check whether an integral can be easily solved using Feynman's approach. What are the main criteria needed to be taken into account?
bpr3003
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$\int\frac{dx}{x-3y}$ when $y(x-y)^2=x$?

If y is a function of x such that $y(x-y)^2=x$ Statement-I: $$\int\frac{dx}{x-3y}=\frac12\log[(x-y)^2-1]$$ Because Statement-II: $$\int\frac{dx}{x-3y}=\log(x-3y)+c$$ Question: Is Statement-I true? Is Statement-II true? Is Statement-II a correct…
RE60K
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Evaluate $\int_2^4\frac{\sqrt{x^2-4}}{x^2}\mathrm dx$

Evaluate $$\int\limits_2^4\frac{\sqrt{x^2-4}}{x^2}\mathrm dx$$ My working: $x=2\sec\theta\quad\Rightarrow\quad\theta=\arccos\left(\frac{2}{x}\right)$ $dx=2\sec\theta\tan\theta…
ahorn
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Easier way to solve $\int\frac{2+\sin x}{\sin x(1+\cos x)}dx$

Is there easier way than universal supstitution to solve this integral $$\int\frac{2+\sin x}{\sin x(1+\cos x)}dx$$?
Meow
  • 1,910
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How do you prove this integral involving the Glaisher–Kinkelin constant

According to wikipedia on the page Glaisher–Kinkelin constant $$\int_0^{1/2} \ln\Gamma(x) dx=\frac32\ln \text{A}+\frac5{24}\ln 2+\frac14\ln\pi$$ I got interested in how you possibly could prove something like that, but couldn't find any citations…
Alice Ryhl
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