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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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Evaluation of triple integral involving floor function

I recently had asked this question that what would be $\int_0^1\int_0^1\int_0^1\left(\lfloor x \rfloor+\lfloor y \rfloor+\lfloor z \rfloor\right) \, dx\,dy \,dz$ and was pretty convinced with the answers saying it's $0$ which was what I guessed…
Nitish
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Can you compute this integral?

I want to integrate $$ \int_0^{\infty}dx\,e^{-ax}\frac{1-(2x)^b}{1-2x} $$ where $a,b>0$. Any ideas about how to proceed?
PhoenixPerson
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Evaluating $\int_0^1 \frac{x^3-1}{\ln x}dx$

The question is to evaluate $$\int_0^1 \frac{x^3-1}{\ln x}dx$$ I tried to put $t=\ln x$ so $dt=\frac1xdx$.Hence I could rewrite the integral as $$\int_{-\infty}^0 \frac{e^{3x}-1}{x}e^xdx$$I couldnot further simplify the integral.Any ideas?Thanks.
Navin
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Integral with absolute value

How to proof that $$ \frac{1}{2}(t^{2H} + s^{2H} -|t-s|^{2H})= H(2H-1) \int_0^t \int_0^s |u-v|^{2H-2} \, du \, dv $$ I have trying to usee derivative in the right hand but I have a doubt as to the born of integration.
Zbigniew
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Is there an easy way to solve this integral?

I just want to see what happens as you increase a with a fixed b (does the ratio increase or decrease). Is there a nice way to do it? $$\frac{\int_a^b x^2 10^{-x} dx}{\int_a^b 10^{-x} dx}$$
Anon123
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Absolute value definition of riemann sum to solve definite integral properties

https://gyazo.com/92d834eb3c6b3c500bda5cf7e2d68fc2 I just used the definition. $$\bigg|\lim_{n\to\infty} \sum_{i=1}^{n} f(x_i^*)\Delta x\bigg| \leq \lim_{n\to\infty} \sum_{i=1}^{n} |f(x_i^*)|\Delta x$$ by def idk what to do. Hints?
user349557
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Why do I get the integral $\int_0^\pi\frac{1}{1-k\cos z}{\rm d}z$ instead of $\int_0^\pi\frac{1}{1+k\cos z}{\rm d}z$?

I have this integral $\int_0^{2\pi}\frac{1}{1+k\cos{x}}{\rm d}x$, which is definitely: $\int_0^{2\pi}\frac{1}{1+k\cos{x}}{\rm d}x = 2\int_0^{\pi}\frac{1}{1+k\cos{x}}{\rm d}x$, but if I do this: $\int_0^{2\pi}\frac{1}{1+k\cos{x}}{\rm d}x =…
atapaka
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Definite integration question $\int_0^{\frac{\pi}{2}} \frac{dx}{3+2\sin x+\cos x}$

Question - $$\int_0^{\frac{\pi}{2}} \frac{dx}{3+2\sin x+\cos x}$$ I have tried this question and also tried to find online on google. But I only found questions that involves only 2 terms. So i dont have any other way except to ask it here. Please…
user404716
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Integrate log and trigonometric function together

Question $$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cos 2x\cdot \log \sin x dx$$ I put $\sin x = t $ Then $\cos x = \sqrt{1-t^2}$ Also $\cos x dx = dt$. But I am getting very large and wrong answer. If you want more information please let me know. Can…
user404716
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How to integrate this question?

Question - $\int_0^{\frac{\pi}{2}} \frac{dx}{1+\cos^2x}$ I put $\cos x = t$ Then $-\sin x\; dx = dt$ But then I am totally confused what to do with $\sin x.$ How to solve this.
user404716
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Find the area of the segment of the parabola $y=x^2-7x+9$ cut off by the line $y=3-2x$.

Please, Sketch the area to make me understand, this question from area bounded curve of integral calculus. Necessary to solve.
user409382
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Copula: How to solve Integral with minimum for computation of Spearmans rho

How can you calculate Spearman's rho of the comonotinicity copula? Comonotonicity Copula: $Cm(u1, u2) = min(u1, u2)$ Spearmans rho: $\rho = 12 \int_{0}^{1}\int_{0}^{1} C(u1,u2) du1 du2 -3$ Obviously, this results in the following: $12…
PalimPalim
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Average width between two curves

Am I right to assume that $$\bar{y} = \frac{{\int_{x_0}^{x_1} f(x) dx - \int_{x_0}^{x_1} g(x) dx}} {x_1 - x_0}$$ where f(x) and g(x) are two polinomials, not crossing each other in this range, and y bar is the average height of the area between the…
user2729400
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How to calculate $\int_{0}^{\frac{\pi}{2}}\frac{\mathrm{d}x}{1+\tan^{a}(x)}$

How to calculate $$ \int\limits_{0}^{\frac{\pi}{2}}\frac{\mathrm{d}x}{1+\tan^{a}(x)} \;, $$ where $a$ is a constant? Any hint will be appreciated.
z3wood
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Determine whether the integral is positive or negative.

I can't think of any proper way to show that this integral is indeed positive (I got that it's positive from Wolfram Alpha). Please, help me!
Sofia.T
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