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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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Tell me if my answer to a question that asks me to use the definition of the definite integral as a limit of sums to compute a integral of a function.

Using the definition of the definite integral as a limit of sums compute $f_{a}^{b} xdx$ for any a,b belonging to the set of real numbers where a
cosygod
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Definite integral of $\sqrt x * \exp \left( \frac{-x}{2\theta} \right)$?

How do I integrate $$ \int^{\infty}_0 \sqrt x * \exp \left( \frac{-x}{2\theta} \right) dx$$ $\theta$ is a strictly positive constant.
Simón Ramírez Amaya
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Definite integral of $|\sin(x)+\sin(nx)|$

I've looked at various online integral calculators including Wolfram Alpha and I can't find anything for absolute values when it comes to sine and cosine waves. For example: If I have $y=|\sin(3x)+\sin(7x)|$, and I want to find the integral of it…
Alice T
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How to solve the definite integral involving greatest integer function?

Let $ [\cdot] $ denote the greatest integer function. Then how to evaluate the integral $$I=\int_0^2 [x^2]dx?$$
DINESH BISWAS
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Does the integral $\int_0^1\sin(1/x)\, \mathrm{d}x$ converge?

I am interested in proving that the integral $$\int_{0^+}^1\sin\left(\frac{1}{x}\right)\,\mathrm d x$$ converges. Can someone show me step by step please?
אורטל כהן
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Is $\int_0^{\pi/2}e^{inx}dx=0$ for $n \neq 0$?

How can we prove this using Cauchy's Integral? We know that $\int_{-\pi}^{\pi}e^{inx} dx = 0$ for $n$ not equal to $0$, and is equal to $2\pi$ for $n=0$, how can we prove that $\int_0^{\pi/2} e^{inx} dx=0$ for $n \neq 0$
Catalan
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Definite integration of product of terms

Finding $$\int^{6}_{0}x(x-1)(x-2)(x-3)(x-4)(x-6)dx$$ My progress so far $$x(x-4)(x-6)(x-1)(x-2)(x-3)=\bigg(x^3-10x^2+24x\bigg)\bigg(x^3-6x^2+11x-6\bigg)$$ How can I find solution Help me
rekha pimoli
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Does $ \int _0^1 \frac{\arctan \left(\frac{x}{x+1}\right)}{\arctan \left(\frac{-2x^2+2x+1}{2}\right)}dx $ exactly equal to $ \frac{1}{2} $?

Is the integral exactly equal to $\frac12$ or is it just an extremely good approximation?
ray lin
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How do I prove that this integral is positive?

Let $f:\mathbb{R}\to\mathbb{R}$ be a monotonic decreasing positive function. While writing a proof, I came across the following integral: $$\int_0^1f(x+n)-f(n+1)dx$$where $n\in\mathbb{N}$. I know that this integral is positive, but how do I write a…
codetalker
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Evaluating a specific definite integral

Can someone help me evaluate the following integral? $$\frac{1}π\int_{-π}^π x^{2n}cos(\frac{nπx}L) dx$$ Integration by parts does not seem to work. $$\frac{1}π\int_{-π}^π x^{2n}cos(\frac{nπx}L) dx =…
LobCity
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my integration variable is canceled to $\int_0^2 1 dx $ - what is that?

i have functions: $f(x)$ and $g(x) = f(x) +1)$. I want to calculate the area between those two functions in $[0;2]$. Therefore, my integral is $\int_0^2 g(x) - f(x) dx $, which results in $\int_0^2 f(x) + 1 - f(x) dx $, and in the end is $f(x)$ cut…
uuu
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Definite integral $\int _{0}^{1}\sqrt [3] {2x^{3}-3x^{2}-x+1}~\mathrm dx$

Evaluate the integral $$\int _{0}^{1}\sqrt [3] {2x^{3}-3x^{2}-x+1}~\mathrm dx.$$
FMath
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Does $\int_{0}^{1} |f(t)|dt=0$ imply that $f(t)=0\ \text{for all } t \in [0,1]$?

$$\int_{0}^{1} |f(t)|dt=0$$ Does this equation imply that $f(t)=0$ for every $t \in [0,1]$? I need a proof of whether the answer is yes or no. I couldn't prove this; I started thinking, if it were zero then the integral is definitely zero, but what…
HHH
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muliple integral question

Calculate the multiple integral : $\int\int\frac{1}{1+x^2}~dA$ where D is the closed triangular region with vertices (0,0), (1,1), and (0,1)? So I have $\int_0^1\int_0^x \frac{1}{1+x^2}~dydx$ which gave me $\frac{ln(2)}{2}$ , but the answer is…
eug
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If the above function is f(x) then f'(x) vanishes at?

$$1+x+\int_1^x\left(\text{ln}^2\left(t\right)+2\text{ln}\left(t\right)\right)dt$$ If the above function is f(x) then f'(x) vanishes at which value of x? I am trying this question by taking the integrating part as z and then I am rewriting f(x)…
DEEPANWITA ROY
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