Mathematics Stack Exchange
Mathematics Stack Exchange

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
4
votes
1 answer

Integrate a function over this set

Problem: if $X$ is an infinite measure subset of $\mathbb R$ and $f:\mathbb R\to\mathbb R$ is a continuous function integrable over $X$ and all its translations and rotations, is $f$ integrable over $\mathbb R$? Trying to find a counterexample, I…
izabera
  • 173
4
votes
1 answer

Different solutions to $\int^\pi_{-\pi} \cos^3(x) \cos(ax)~dx$

Linked to my previous question, when solving the following integral ($a$ is an integer) I get: $$\int^\pi_{-\pi} \cos^3(x) \cos(ax)~dx = \frac{2a(a^2-7)\sin(\pi a)}{a^4 - 10a^2 + 9}$$ However, trivially, $\sin(\pi a) = 0$ for all integer values of…
H G
  • 41
4
votes
2 answers

How to compute the integral $\int _{0}^{1}{\tan^{-1}\left(x\right) \over {1+x}}dx$

How to compute the integral $$\int_0^1 {\tan^{-1}\left(x\right) \over {1+x}}\,{\rm d}x$$
DebojyotiMukh
  • 915
4
votes
1 answer

Definite integral involving square root of polynomial

How can I solve the integral below? $$\int_0^1 (x^4+x^3+x^2+x^1+1)^{1/2} \mathrm dx $$
Mahsa Ghaffari
  • 103
4
votes
1 answer

Elementary function solution of $\int_0^{\varepsilon} \sqrt{\varepsilon^2-y^2}\,\frac{{\rm artanh}\,y}{1-y^2}{\rm d}y$

I would like to to know if the following integral can be done in terms of elementary functions of $\varepsilon$ $$ I(\varepsilon)=\int_0^{\varepsilon} \sqrt{\varepsilon^2-y^2}\,\frac{{\rm artanh}\,y}{1-y^2}{\rm d}y\,. $$ I tried writing the ${\rm…
user12588
  • 399
4
votes
0 answers

Explicit form for these logarithmic integrals

Can either of the following integrals be expressed in explicit forms in terms of known constants? They can certainly be expressed in terms of special values of hypergeometric functions with many parameters, but I am interested in expressions in…
SearchingForPeriods
  • 41
4
votes
1 answer

Calculate integral using the value of another one

Calculate the integral $\int_{-1}^{3} t^3(4 + t^3)^{-1/2} dt$, given that $\int_{-1}^{3} (4 + t^3)^{1/2} dt = 11.35$. Leave the result in function of $√3 \text{ and }√31$. I have tried $\int_{-1}^{3} (4 + t^3)^{1/2} dt = 11.35.$ $\int_{-1}^{3}…
Heyner Márquez
  • 51
4
votes
1 answer

For what value of $a$, $\int_{0}^{2}a^xdx=3$?

For what value of $'a'$, $\int_{0}^{2}a^xdx=3$? I solved further and got $$\frac{a^2-1}{\ln(a)}=3\Rightarrow a^2-1=3\ln(a)$$ $\forall a\ne1$. Putting this in wolfram alpha gives $a=1.464...$Is there any way to get this without using wolfram alpha?
Sam
  • 41
4
votes
1 answer

Integral in two variables

I have to prove that $$\int_D xy\; dxdy=\frac{3}{8}$$ with $D=\{(x,y)\in \mathbb{R}^2: x^2+y^2 \geq 1,\ \frac{x^2}{4}+y^2 \leq 1,\ x \geq 0,\ y \geq 0 \}$. So I have the intersection of ellipse $\frac{x^2}{4}+y^2 \leq 1$ and the ball $x^2+y^2 \leq…
Mario
  • 717
4
votes
1 answer

Calculating $\int_0^1 \frac{x\ln(x+1)}{x^2+1} dx$ without using complex numbers

One can find the antiderivate with help of the partial fraction method introducing complex numbers: $$\frac{x\ln(x+1)}{x^2+1}=\frac{1}{2}\left(\frac{\ln(x+1)}{x-i}+\frac{\ln(x+1)}{x+i}\right).$$ The result is a complicated function containing…
Ben Hur
  • 151
4
votes
5 answers

A "tricky" integral: $\int_0^{\infty} t e^{-nct} (1-e^{-ct})^m dt$

In an article in the current (May 2013) issue of the College Mathematics Journal, they say that the following integral is "tricky to evaluate": $\int_0^{\infty} t e^{-nct} (1-e^{-ct})^m dt$ where $n$ and $m$ and n are non-negative integers and $c$…
marty cohen
  • 107,799
4
votes
2 answers

Show $\int_0^b \sqrt{\frac{1 + \cos \theta}{\cos \theta - \cos b}} d\theta= \pi$ for $0 < b < \pi.$

I came across this integral in evaluating the time a particle takes to travel between points on the cycloid. I was able to use substitution to do the integral in the case $b = \pi/2.$ I have not not been able to show the result for other values of…
abel
  • 29,170
4
votes
1 answer

Area bounded by hyperbola

I tried to find the area of the reactangular hyperbola $$xy=1$$ from $x =0$ to any arbitrary $x$ by using integrals,I found the area to be infinity whereas when I found the area of $$y=\ln(x)$$ from zero to let's say $x=1$ then I found the area to…
Nalin Yadav
  • 97
4
votes
0 answers

Proving $\int_{0}^{\pi/2}(\log(\sin x))^2dx = \frac{1}{24}\cdot(\pi^3 + 12\pi(\log 2)^2)$

Proving $$\int_{0}^{\pi/2}(\log(\sin x))^2dx = \frac{1}{24}\cdot(\pi^3 + 12\pi(\log 2)^2)$$ without making use of gamma function,digamma function, hypergeometric function.
Godly Light
  • 151
4
votes
2 answers

How to evaluate $\int_{0}^{\infty}\ln^2(x)\ln(1+x)\ln^2\left(1+\frac{1}{x}\right)\frac{dx}{x}$

$$I=\int_{0}^{\infty}\ln^2(x)\ln(1+x)\ln^2\left(1+\frac{1}{x}\right)\frac{\mathrm…
user569129
1 2 3
…
76 77