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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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Advanced convolution u-substitution involving error functions

In finishing the evaluation of a partial differential equation, I've arrived at a stage of a convultion integral I'm stuck at. I have to evaluate the following integral $\frac{a}{\sqrt{\pi}}\int^{t}_0 \frac{k}{\sqrt{t-\tau}}\exp({\frac{−x^2}{4a^2…
slamDuncan
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Integral of a case function

Assume the following function, with $x, a, b \in \! \mathbb{R}$ $$ f(x,a,b) = \begin{cases} x+a+b & \mbox{for } ~ a-b \le x \le a+b \\ 0 & \mbox{elsewhere} \end{cases} $$ How can the following integral be computed? $$ I(x) =…
thekyria
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How to evaluate this definite integral which involves sine

The input signal for a given electronic circuit is a function of time $V_{in}(t)$. The output signal is given by $$V_{out}(t) = \int_0^t \sin(t-s)V_{in}(s)ds$$ Find $V_{out}(t)$ if $V_{in}(t) = \sin(at)$ where $a > 0$ is some constant I believe…
Alex
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How to evaluate $\int^{+\infty}_0 e^{−ax^2}\cos(bx)\,dx=\sqrt{\frac{\pi}{2a}}e^{−b^2/4a}$

I've been at this for a few hours now, and it's frighteningly similar to the problem stated here: How to prove $\int_0^\infty e^{-x^2}cos(2bx) dx = \frac{\sqrt{\pi}}{2} e^{-b^2}$ but with enough change that it's still proving problematic. I also…
slamDuncan
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Evaluating the integral : $\int_{1}^{2}\frac{x+\tan x}{x+\sin x}dx$

$Q.$ Evaluate the following integral : $\int_{1}^{2}\frac{x+\tan x}{x+\sin x}dx$. Numerically I found that the answer is roughly $1.000006$ but I am unable to compute using the analytic methods. I tried first computing by splitting: …
creative
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Evaluate $\frac{1}{\tau} \int\limits_0^\tau{{(e^{\alpha t}-1)e^{-\beta t}}}dt$

I have to evaluate $\frac{1}{\tau} \int\limits_0^\tau{{(e^{\alpha t}-1)e^{-\beta t}}}dt$ , where ατ=1. I did he following: But they have the following answer: Have I forgotten something? I did it multiple times. Can't get nearer to their result.…
Tanatos Daniel
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Gaussian integral of a function with nonzero mean

From the wikipedia article, for a Gaussian integral of an analytic function we have that I'm trying to obtain a similar formula when there is a linear term in the Gaussian (ie the Gaussian has a nonzero mean). My guess is that $$ \int f(x) \exp…
psmith
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Express the limit as a definite integral on the given interval [Difficulty Interpreting Question]

I understand that they want the integral as an answer, but how would I go about getting this. Sorry this is all just very new to me and i just want to grasp the concepts :)
JoshBoiskin
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Definite integrals

$$\int_0^{1.5}[x^2]dx$$where [.] denotes the greatest integer function, is equal to : (1) $\sqrt{2}-2$ (2) $2 –\sqrt{2}$ (3) $2 + \sqrt{2}$ (4) None of these What I did, I broke the function into two parts..one with limits from 0 to 1.the problem…
Yash Lekhwani
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A certain class of integrals: substitution possibilities?

I am considering integrals of the form $$ I_{n,m}=\int_{-y_0}^{y_0}y^m\left(1-\frac{y^2}{y_0^2}\right)^{\frac{n}{2}}\text{d}y, $$ where $m\in \mathbb{N}_{>0}$ and $n=2k+1$, with $k\in\mathbb{N}$. Is there a simple substitution to solve these? At…
Funzies
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length along function

I am no specialist in integration but I would like to know what is the (arc?) length of the function (along the function) between $f(a)$ and $f(b)$ where $ f(x) = \sqrt{(x^2+1)} $ I would like to see the deduction.
Willemien
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Definite Integral of $1+\sqrt{9-x^2}$?

Right, so I'm to find the definite integral (interpreting it as an area)... $\int^0_{-3}(1+\sqrt{9-x^2})dx$ How do I go about doing this? I am to specifically use the following theorem to work it out... If $f$ is integrable on $[a,b]$…
Wolff
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Definite Integral $\int_{-\infty}^{\infty}\frac{x\sin x}{(x^2+a^2)(x^2+b^2)}\,\mathrm{d}x$

$$\int_{-\infty}^{\infty}\frac{x\sin x}{(x^2+a^2)(x^2+b^2)}\,\mathrm{d}x$$ This is easy to evaluate with complex analysis but is there an elementary way (substitution, partial fractions, integration by parts)?
user85798
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Integrating sinc / gaussian function with 2nd order polynomials as arguments

I am wondering how to integrate $$ I(a, b) = \int_{-\infty}^{\infty}dx \frac{\sin(x^2 + a)}{x^2 + a} \exp(i[x-b]^2) $$ Thus far I tried integration by parts and contour integration but could not find a solution. This integral arises in connection…
user140462
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question about fourier coefficient

I got this on my physics class but I post it here since it relate to math more Here, the explanation is $$s(t)e^{-j\frac{2\pi lt}{T}}=\sum c_ke^{j\frac{2\pi kt}{T}}e^{-j\frac{2\pi lt}{T}} \\ \implies\int s(t)e^{-j\frac{2\pi lt}{T}}=\int\sum…
aukxn
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