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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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Prove $|\int_{n}^{n+p} \sin (x^2)dx|\leq 1/n$ where $p>0$.

Prove $$\left|\int_{n}^{n+p} \sin (x^2)dx\right|\leq \frac{1}{n}$$ where $p>0$. Maybe, we can improve namely enhance to prove $$\left|\int_{n}^{+\infty} \sin (x^2)dx\right|\leq \frac{1}{n},$$ which is hold?
mengdie1982
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Dummy Variables in Definite Integration.

The variable appearing in a definite integral is known as a dummy variable, which means $\int_a^b f(x)dx$ is the same thing as $\int_a^b f(t)dt$. Now this result comes from the fact that we are simply substituting $x=t$. If we substitute anything…
Anurag Saha
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SOLVED! Evaluating integral of e and hyperbolic functions

This is the only question in my homework that I can't finish. I'm having issues with taking the integral of the function. I have set $e^{x}$ as the u for substitution and also $sinh(x)+cosh(x)$, too but nothing worked. $\int_{-12}^{12}…
binibiningtinamoran
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If $f$ is a continuous function on $\Bbb{R}$, then prove that $\int_0^1 f(x)x^2 \, \mathrm{d}x = f(c)/3$ for some $c \in[0,1]$.

I tried to solve this using Riemann sum. I get integral is $f(c)c^2$ for some $c$ in $[0,1]$. But I couldn't show that it is $f(k)/3$ for some $k$ in $[0,1]$. Could you help me? Thanks.
Balasubramannyan S
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Finding an integral's max and min values

So I have tried to solve this question in two methods: Given that $g\leqslant\int_{0}^{1} \frac{x}{x^2 +1} dx \leqslant h$, Find g, h. The first way I tried is defining $f(x) = \frac{x}{x^2+1}$ and then I found its maximum and minimum values using…
passepartout
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find the minimum value of "a" such that the following integral holds good, [ ] represents the greatest integer function

Finding minimum value of $a$ such that following integrals holds $$\int^{a}_{0}\bigg\lfloor \tan^{-1}(\sqrt{x})\bigg\rfloor dx=\int^{a}_{0}\bigg\lfloor \cot^{-1}(\sqrt{x})\bigg\rfloor dx$$
anairudh jm
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Decide if the following improper integral converges. If so, calculate value

Show that $$\int_1^2\cfrac {dx}{x\ln^2 x} \qquad \qquad \qquad (a)$$ Converges or diverges , I can think of using a substitution, $u=\ln x$ , $du=\cfrac 1xdx$ \begin{align} & = \int_1^2\cfrac {dx}{x\ln^2 x}\\ & = \int_1^2\cfrac {du}{u^2}\\ & = …
Sebastian Fernandez
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Applying Fundamental Theorem of Calculus on multiplication of functions

I gotta differentiate this, and I don't know if my answer is correct $$ F(x) = \int_0^{f(x)} f(u)g(u) \, du = $$ Should it be $$F'(x) = f(f(x)) f'(x) g(f(x)) f'(x)$$ And if it is or not correct, could you please explain why?
Max Villegas
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Just want to double-check my answer to a volume bounded by a plane and a surface

Can someone please spend a couple minutes in checking my answer, it will be much appreciated Question Find the finite volume bounded by $y=1-x^2-4z^2$ and $y=0$. My answer The answer I got was $\frac{\pi}{4}$. If there is a difference between my…
UnsinkableSam
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Find the derivative of g(x) = $\int_{\sin x}^{\cos x} {{t+\sin t^5}\over1+t^2}dt$

$$\int_{\sin x}^{\cos x} {{t+\sin t^5}\over1+t^2}dt$$ I would use formula : $${d\over dx}\int_{a}^{x}f(t)dt = f(x)$$ but unfortunately I don't know how to find $sinx$ and $cosx$ in order to get a constant 'a'.
retne
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If $J_{m, n}=\int_{0}^{\pi/2}\cos^mx\cos nx\;dx$, then evaluate $\frac{J_{6,3}}{J_{5,2}}$.

If $\displaystyle J_{m, n}=\int_{0}^{\pi/2}\cos^mx\cos nx\;dx$, where $m$ and $n$ are natural numbers, then evaluate $\dfrac{J_{6,3}}{J_{5,2}}$. My try: $$J_{m,n} = \int^{\pi/2}_{0}\cos^m x \cos nx dx = \frac {\cos^m x \sin nx}{m +…
DXT
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Definite integral of a given function.

How can I compute definite integral of the following function? $\int_{x(0)}^{0} \frac{dx}{k_2\,\sin x + k_1\frac{\cos x - 1}{\sin 2x}}$ $k_1$ and $k_2$ are positive constants. At this point, I know that the function is odd. How should I approach the…
Scholar
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Minimum of parameter integral

Find the minimum value of the function $f: R \to R, f(x)= \int_{0}^{1} {|x-t|}^3dt$ I computed the function analysing 3 cases: $x \leq 1, x \in (0,1), x \geq 1 $ And then i studied the extrem with derivatives obtaining that f has $1/2$ as a minimum…
amarius8312
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How to compute $\int_{a}^{b}\sqrt{\mathrm dx}$

How to compute $\int_{a}^{b}\sqrt{\mathrm dx}$? I do not think setting $\sqrt {\mathrm dx} = \mathrm du$ is the right idea. Is it even sensible to have $\mathrm dx$ square rooted?
Rob
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How to find the closed form of a multiple integral

I want to find the closed form of the following $$\int\limits_{t
xuce1234
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