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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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Definite integral, $\frac 1{\ln(x)}$

What is $$\int_0^{\pi^{2}}\frac 1{\ln(x)}dx$$ I tried using complex residues and some identities, but no luck. Any suggestions?
Tdonut
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How to evaluate the integral $\int_1^{\infty}[u^\alpha-(u-1)^{\alpha}]^2du$?

The integral \begin{equation} I=\int_1^{\infty}[u^\alpha-(u-1)^{\alpha}]^2du,\quad -1/2<\alpha<1/2 \end{equation} comes from the stochastic process fractional Brownian motion. Can someone show how to evaluate it? Thank…
ecook
  • 399
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How to calculate the integral $\int_{-\infty}^{\infty} \frac{dx}{(\exp(x)- x)^2 + \pi^2} = \frac{1}{1 + W(1)}$

On Mathworld one finds without proof the integral $$\int_{-\infty}^{\infty} \frac{dx}{(\exp(x) - x)^2 + \pi^2} = \frac{1}{1 + W(1)}$$ where $W$ denotes the Lambert W function. How can one show this? The link given on Mathworld is broken.
user111187
  • 5,856
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How to do integral $\int_0^T \frac{1}{t\sqrt{t(T-t)}}e^{-\frac{b^2}{2t}}dt$ and $\int_0^T \frac{1}{\sqrt{t(T-t)}}e^{-\frac{b^2}{2t}}dt$?

I met these two integrals but don't know how to do them: $$I_1 = \int_0^T \frac{1}{t\sqrt{t(T-t)}}e^{-\frac{b^2}{2t}}\text{d}t$$ $$I_2 = \int_0^T \frac{1}{\sqrt{t(T-t)}}e^{-\frac{b^2}{2t}}\text{d}t$$ where $b>0$, $T>0$. Please kindly help? Thanks…
athos
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Find the value of the integral $\int_0^{2\pi}\ln|a+b\sin x|dx$ where $0\lt a\lt b$

Find the value of the integral $$\int_0^{2\pi}\ln|a+b\sin x|dx$$ where $0\lt a\lt b$. What is the use of this inequality. I tried to integrate the integral by parts, but the integral of the 2nd term was quite messy.Please help.
User56192
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Multiple integral of min

How can this be proven? $$ \int_0^1 \int_0^1...\int_0^1 min(x_1,x_2,...,x_n) dx_1dx_2...dx_n = \frac1{n+1} $$ I tried to split the last integral in $$\int_0^{min(x_1,x_2,...,x_{n-1})} x_ndx_n + \int_{min(x_1,x_2,...,x_{n-1})}^1…
user42768
  • 753
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Help solving this integral $\int_{-\infty}^{\infty} \frac{x^2 e^{-x^2/2}}{a+bx^2}dx$

So, I've got an integral in the following form: $$\int_{-\infty}^{\infty} \frac{x^2 e^{-x^2/2}}{a+bx^2}dx$$ where $b<0$ and $a\in\mathbb{R}$. I've tried substituting $y=x^2$ (after changing changing lower limit to 0 and multiplying by 2 of course)…
M.B.M.
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How to prove the equality $\int^{na}_{ma}\frac{\ln(x-a)}{x^2+a^2}\,dx = \int^{\frac{a}{m}}_{\frac{a}{n}}\frac{\ln(x+a)}{x^2+a^2}\,dx$?

How to prove the equality $$\int^{na}_{ma}\frac{\ln(x-a)}{x^2+a^2} \, dx= \int^{\frac{a}{m}}_{\frac{a}{n}}\frac{\ln(x+a)}{x^2+a^2}\, dx,$$ where $a, m, n$ are strictly positive numbers such that $mn=m+n+1$?
medicu
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Determine antiderivative of $f$

So I have a function: $$f (t) = \begin{cases} -1, & \text{if } {-2} \le t \le 0 \\ 2 - \sqrt{t^2-6t+9}, & \text{if } 0 < t < 6 \\ -1, & \text{if } 6 \le t \le 8 \\ \end{cases}$$ I need to find the antiderivative of $f(t)$ on the interval: $-2 < t <…
PiotreX
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Calculate the value of $I(9)/I(3)$ when $I(m)=\int_{0}^{\pi}\ln\left(1-2m\cos(x)+m^2\right)\,dx$

We are given that $$I(m)=\int_{0}^{\pi} \ln\left(1-2m\cos (x)+m^2\right)\,dx.$$ I could see that there weren't any standard techniques to calculate this integral directly so I concluded that there must be some kind of reduction formula to be…
infinite-blank-
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Trigonometric and Exponential Integration

If $\displaystyle I_{n} =\int^{\infty}_{\frac{\pi}{2}}e^{-x}\cos^{n}(x)dx.$ Then $\displaystyle \frac{I_{2018}}{I_{2016}}$ is Try: using by parts $$I_{n}=\int^{\infty}_{\frac{\pi}{2}}e^{-x}\cos^{n}(x)dx$$ $$I_{n}=-\cos^{n}(x)\cdot…
DXT
  • 11,241
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4 answers

series sum $\frac{1}{2}\cdot \frac{1}{2}k^2+\frac{1}{2}\cdot \frac{3}{4}\cdot k^4+\frac{1}{2}\cdot \frac{3}{4}\cdot \frac{5}{6}\cdot k^6+\cdots $

Finding value of $\displaystyle \int^{\pi}_{0}\ln(1+k\cos x)dx$ for $0
jacky
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Finding $\int^{\pi}_{0}f^{-1}(x)\,\mathrm dx$

$\def\d{\mathrm{d}}$If $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(x)=x+\cos x$, find$$\int^{\pi}_{0}f^{-1}(x) \,\d x.$$ Try: put $x=f(t)$ and $\d x=f'(t) \,\d t$, so \begin{align*} \int^{f^{-1}(\pi)}_{f^{-1}(0)}tf'(t) \,\d t…
DXT
  • 11,241
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Prove: there is a point where the derivative is zero

Let $f:[-5,3]\to \mathbb{R}$ such that $f$ is differentiable, moreover $\int_{-5}^{-1}fdx=\int_{-1}^{3}fdx$ Prove: there is $x_0\in[-5,3]$ such that $f'(x_0)=0$ $f$ is differentiable $\Rightarrow$ continuous$\Rightarrow$ integrable, now,…
gbox
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$\displaystyle\int_{0}^{\infty}\left[ne^{-x}\right]dx=\log\frac{n^{n-1}}{(n-1)!}$

$\displaystyle\int_{0}^{\infty}\left[ne^{-x}\right]dx=\log\frac{n^{n-1}}{(n-1)!}, \ $ where $[\,\cdot\,]$ is the greatest integer function. Put $e^{-x}=t$ $$ \begin{aligned} \int_{0}^{\infty}\left[ne^{-x}\right]dx & =\int_{1}^{0}\left(\left[n…
diya
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