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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.

20559 questions
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Closed form for $\int_0^1 \frac{x^a dx}{(1+x^b)^c}$.

By means of a substitution, the integral can be reduced to $$\int_0^1 \frac{x^p dx}{(1+x)^q}$$ but no method that I've thusfar tried seems tenable. This Beta-like integral cannot be reduced into the Beta function (i.e. $\int_0^\infty \frac{x^p…
Meow
  • 6,353
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Arc length of $\sin^3({\frac x3})$ from $0$ to $6\pi$

How can I find the arc length for $P(x) = \sin^3\left({\frac x3}\right)$ from $0$ to $6\pi$. I try to find: $$\int_0^{6\pi} \sqrt{1 + \cos^2\left({\frac x3}\right)\sin^4\left({\frac x3}\right)}\,\mathrm dx$$ but I can't. Help me please, or just…
Alexander
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Integral of the function $S(x)=\ln\left(1-\frac{x}{\exp(x)}\right)$

I have to check if the following series: $$S(x)=\sum_{k=1}^{\infty}\frac{x^k}{k\exp(kx)}$$ gives a function of $x$ $$S(x)=-\ln\left(1-\frac{x}{\exp(x)}\right)$$ for which: $$J=\left|\int_{0}^{+\infty}S(x)dx\right|\lt\infty$$ I used Maple and…
Riccardo.Alestra
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Evaluating an indefinite integral

I need help evaluating the following indefinite integral explicitly $$\int \frac{1}{1+t^{2^{-n}}} dt$$ I would appreciate any help
Ethan Splaver
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Evaluating the integral of the square of an infinite sum

I want to determine the value of $s$ for which a particular equation gives the smallest integral with respect to $a$ over the interval $[0,1]$. The three main terms in the equation each involve infinite sums, so the summations are (at least…
CrimsonDark
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Prove that $f(x)=\int_{0}^x \cos^4 t\, dt\implies f(x+\pi)=f(x)+f(\pi)$

How to prove that if $f(x)=\int_{0}^x \cos^4 t\, dt$ then $f(x+\pi)$ equals to $f(x)+f(\pi)$ I thought to first integrate $\cos^4 t$ but this might be the problem of Definite Integral
DebojyotiMukh
  • 915
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The different results of the definite integral by two methods

\begin{array}{l} Find\ the\ value\ of\int _{\frac{\pi }{2}}^{\frac{3\pi }{2}}\sqrt{( 1+sin\theta )^{2} +cos( \theta )^{2}} d\theta \\ \\ Method\ 1\ :\ Substitution\\ \int \sqrt{( 1+sin\theta )^{2} +cos( \theta )^{2}} d\theta \\ =\int…
rann rann
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Definite integral of $\sqrt{\log\left(1+\frac{1}{2x}\right)}$

Can someone explain to me how to solve the integral \begin{align*} \int_0^1 \sqrt{\log\left(\frac{1}{2x}+1\right)}\mathrm{d} x \end{align*} I know it is finite by simulation in R fun <- function(x) sqrt(log(1+1/(2*x))) integrate(fun, 0, 1) 0.9304908…
wonderer
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$\int_{-\infty}^{+\infty} dx \, \frac{e^{-x^2}}{x-i} $

I want to know how to compute the following integral: $$\int_{-\infty}^{+\infty} dx \, \frac{e^{-x^2}}{x-i} \, .$$ Mathematica gives $$i \, e \, \pi \, \text{erfc(1)} \, ,$$ but I don't know how to calculate it. I think contour integral does not…
MBolin
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How to evaluate an integral that comes out as undefined

How do I evaluate $\int_0^{\pi}\frac{1}{5+3\cos{x}}dx$? Because for this type of integral I would normally use the substitution $t=\tan{\frac{x}{2}}$ but $\tan{\frac{\pi}{2}}$ is undefined. How can I evaluate this integral still using this method (I…
Pen and Paper
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How to prove that $ \int_0^{\infty } \frac{\psi ^{(0)}(z+1)+\gamma }{z^{15/8}} \, dz = \frac{2 \pi}{\sqrt{2-\sqrt{2}}} \zeta(\frac{15}{8}) $?

This integral is from an exercise in an old calc book by Edwards from the 20s. The $\psi$ is the Digamma function and the $\zeta$ is the Riemann Zeta function. How would one go about trying to prove this? Does this require just basic substitution?…
DecarbonatedOdes
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What are some Integration Techniques unrelated to the Antiderivative?

Are there any definite integration techniques which I could learn (calc AB student)? I mean techniques which don't require you to find the anti derivative. Thanks!
Ovi
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Following integral?

How to solve the following: $$\int_1^{x} \lfloor t\rfloor dt $$ I can conclude the answer is asymptotic to $\displaystyle \frac{1}{2} x^2 - \frac{1}{2} x$ and specifically it looks just like $\displaystyle \frac{1}{2}x^2$ except entirely linear…
Sidharth Ghoshal
  • 16,771
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How to evaluate $\int_{-\pi/2}^{\pi/2} \tan x \cos (A \cos x +B \sin x) \, dx$?

$$\int_{-\pi/2}^{\pi/2} \tan x \cos (A \cos x +B \sin x) \, dx$$ Is it possible to calculate this? Both A and B are non-zero and assumed to be real numbers. I tried Integrate[Tan[x]*Cos[A*Cos[x]+B*Sin[x]],{x,-Pi/2,Pi/2},PrincipalValue->True], but it…
Ui-Jin Kwon
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Find $\int_{\frac{1}{4}}^4\frac{(x+1)\arctan x}{x\sqrt{x^2+1}} dx$

I have to find this integral: $$\int_{\frac{1}{4}}^4\frac{(x+1)\arctan x}{x\sqrt{x^2+1}} dx$$ My attempt was to split the integral: $$\int_{\frac{1}{4}}^4\frac{(x+1)\arctan x}{x\sqrt{x^2+1}}=\int_{\frac{1}{4}}^4\frac{x\arctan…
user754302
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