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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.

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Evaluating $\frac{1}{\pi} \int_a^b \frac{1}{\sqrt{x(1-x)}}dx$

I can't calculate the following integral: $$\frac{1}{\pi} \int_a^b \frac{1}{\sqrt{x(1-x)}}dx,$$ where $[a, b] \subset [0,1]$. Can someone, please, give me a hint? Thank you!
g.pomegranate
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Find the value of integral $\int^{2\pi}_{-\frac{\pi}{2}}\lfloor \cot^{-1}(x)\rfloor dx$

Find the value of integral $$\int^{2\pi}_{-\frac{\pi}{2}}\lfloor \cot^{-1}(x)\rfloor dx$$ $\lfloor x \rfloor = x-\{x\}$ and $0\leq \{x\}<1$ for $0\leq x<\cot(1),\lfloor \cot^{-1}(x)\rfloor = 0$ and for $\cot (1)\leq x<2\pi,\lfloor…
DXT
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Evaluation of $\int^{\infty}_{0}e^{-ax}(\cos x)^ndx$

Evaluation of $$\int^{\infty}_{0}e^{-ax}(\cos x)^ndx,a>0$$ $\bf{My\; Try::}$ Let $$I = \int^{\infty}_{0}e^{-ax}(\cos x)^ndx\;,$$ Now Using euler substution $$\cos x = \frac{e^{-ix}+e^{-ix}}{2}$$ So $$I =…
juantheron
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Evaluate the definite integral: $\int\limits_0^{\pi/2}\frac{\sin x-\cos x}{\sqrt{1-\sin 2x}}\, dx$

I would like to evaluate : $$\int\limits_0^{\pi/2}\frac{\sin x-\cos x}{\sqrt{1-\sin 2x}}\, dx$$ progress $I=\int\limits_0^{\pi/2}\frac{\sin x-\cos x}{\sqrt{1-\sin 2x}}\, dx$ $=\int\limits_0^{\pi/2}\frac{\sin x-\cos x}{\sqrt{(\sin x-\cos x)^2}}\,…
user356595
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I need to find the following definite integral

Let $f(x)= e^x + 2x + 1$ and $g(x)$ be inverse of $f(x)$. Find $\int_0^{e+3}g(x) dx$ My attempt: I tried to find the inverse of $f(x)$ but was not able to do it. Then I switched to method of area. But don't know how to proceed. I know that if $A$ is…
Vidit Kulshreshtha
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Absolute Value Integral

Evaluate $$\int_{0.1}^{1}|(\pi)( x^{-2})sin(\pi \cdot x^{-1})|dx$$ The above has to be computed without a calculator. I know that $$\frac{d}{dx}[cos(\pi \cdot x^{-1})] = (\pi)( x^{-2})sin(\pi \cdot x^{-1})$$ Applying the limits to the left hand…
StopReadingThisUsername
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Determining correct value given two definite integrals

Let $f(x) = \int_1 ^x g(k) dk$ and $g(t) = \int_0 ^{4\tan(t)} \sqrt{ 16 + w² } dw$ Determine the correct value of $f''(\frac{\pi}{4})$. I know how to solve this by solving for $g(t)$. Then I know that $f'(x) = g(x)$; $f''(x) = g'(x)$ and I get my…
Luis
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Raise part of the integrand to positive power and determine the sign of the integral

Given function $h_1(s)\ge0, h_1'(s)\ge0$, $h_2(s)\ge0, h_2'(s)\ge0$ for $s\in[0,\bar s]$ and $g(s)$ is a density function on $[0,\bar s]$. I already shown that $$ \int_0^{\bar s}h_1(s)g(s)ds\ge \int_0^{\bar s}h_2(s)g(s)ds $$ and $$ \int_0^{\bar…
Glenn
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Different Solution for $\int_{0}^{\pi/2}\frac{1}{1+\tan(x)^{\sqrt{2}}}$

I Believe that I have the optimal solution, using the fact that:…
Andres Mejia
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Some guesses and questions about $\int_a^{\infty}\:f(x)dx$

$\int_a^{\infty}\:f(x)dx$ is convergent, $f(x)$ is monotonically decreasing and continuous. Consider: Is $xf(x)$ also a monotonic decreasing function?prove it or give a counterexample, please. Besides, Is $\lim_{x\to +\infty} x\ln(x) f(x)=0$…
user350652
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evaluation of this exponential integral?

how could i evaluate $$ \int_{0}^{a(E)}\sqrt{E-16\pi ^{2}e^{4x}} $$ where 'a' is the point so $ E-16\pi^{2}e^{4a}=0 $ this appear s in Quantum mechanic so i think the answer is something like $$ E^{1/2}log(E) $$ with some constants but it goes like…
Jose Garcia
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how to get the function(s) under the integral sign in definite integral

Say we have the definite integral: $$\int_a^b{f(x)\, \mathrm{d}x} = \alpha$$ Given, $a, b,$ and $\alpha \in \mathbb{R}$, is it possible to get the functions $f(x)$ in general case? Thank you
user265759
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Integration in a disk in 2D plane

I want to ask how to integrate $\int_{B_r(0)} \dfrac{1}{\sqrt{|x|}|x-a|^2}dx$ where $x \in \mathbb{R}^2$, does it has a explicit solution? or some estimate? Thank you!!
mnmn1993
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What will be the value of $h'(1)?$

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a continuous function.Define $h:\mathbb{R}\rightarrow\mathbb{R}$ by, $$h(x)=\int_0^x\int_0^xf(u,v)dudv$$ What will be the value of $h'(1)?$ I am getting no clue about this problem.I tried but don't know…
P.B.
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Find Time Traveled with Given Distance and Linear Acceleration

A sprinter passes the 10 meter mark as 2 seconds pass since the race started. Find the top speed and acceleration of the sprinter during the interval from when the race started to the time he reached the 2 second mark. I don't understand why…
Saketh Malyala
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