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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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How to integrate $-\int_{-\pi/2}^{\pi/2}{\frac{\sin x}{1+e^x}dx}$?

I need help with this integral: $$-\int_{-\pi/2}^{\pi/2}{\frac{\sin x}{1+e^x}dx}$$ I know this might feel like me asking you to do my homework, but it isn't. I'm simply stuck in this problem and can't figure how to solve it even after investing a…
Utkarsh Verma
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Analysis Riemann integral

Let $f$ and $g$ be continuous functions on $[a,b]$ and $g>0$. Show that there exists a $c \in [a,b]$ such that $$\int_a^b f(x)g(x) \, \mathrm{d}x = f(c)\int_a^b g(x) \, \mathrm{d}x.$$
yamita kumari
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Calculate the finite value of $E(\left|X-Y\right|)$ where $X$ and $Y$ are standard uniform random variables

Assume that $X$ and $Y$ are uniform random variables of the uniform distribution on $[0, 1]$. Then $E(X) = 0.5$, since the $E(X)$ is the mean which is $$ E(X) = \frac{1+0}{2} $$ My intuitive assumption for $E(\left|X - Y\right|)$ was $0$, since…
devssh
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Integral involving $u^{-u}$

I'm trying to solve the following definite integral $$\int_0^{\infty}u^{-u+a}\,du$$ with $a>0$. Using integration by parts I've arrived $$\int_0^{\infty}u^{-u+a}\,du=\frac{1}{1+a}\int_0^{\infty}u^{-u+a+1}(1+\ln(u))\,du$$ which is a worst…
popi
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Definite integration inequality

I tried doing it by area concept. Since the function varies from 1/2 to 1 in 0 to 1 it's area should also vary from 1/2 to 1 in 0 to 1 and similarly for 1 to 2 as well. When I add the answer should be from 1/2 to 3/2 but it is not in the option and…
Akash Gautama
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Where does the $\pi$ comes from?

I have calculated with Mathematica the integral: $$\int_{-h/2}^{h/2}dz\int_{-R}^{R}dx\int_{-\sqrt{R^2-x^2}}^{\sqrt{R^2-x^2}}dy(x^2+z^2)$$ the result is: $$\dfrac{\pi}{12}h^3R^2+\dfrac{\pi}{4}hR^4$$ I am surprised about the $\pi$: where does it comes…
mattiav27
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Integral with sine inside log

I want to evaluate the value of the following integral: $$\int_0^\frac{\pi}{2}\frac{1}{\sin^2{x}}\ln{\frac{1+a*\sin^2{x}}{1-a*\sin^2{x}}}dx$$ i have tried 2 methods, but fail to…
Reynan Henry
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product of $2$ definite integrals

If $\displaystyle I=\int^{\infty}_{0}e^{-2x}\cdot x^6dx$ and $\displaystyle J=\int^{2}_{0}x(8-x^3)^{\frac{1}{3}}dx$ Then product of $I$ and $J$ equals Try: For $\displaystyle I = \int^{\infty}_{0}e^{-2x}\cdot x^6dx =…
DXT
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How to find limits of this volume integration?

The question is If $\vec { F } = x \hat { i } + y \hat { j } + z \hat { k }$ then find the value of $\int \int _ { S } \vec { F } \cdot \hat { n } d s$ where $S$ is the sphere $\{(x,y,z)\in\mathbb{R}^3 \vert x ^ { 2 } + y ^ { 2 } + z ^ { 2 } =…
Hawkingo
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Integrals of generally continuous functions: definition and example

$f(x)$ is called "generally continuous functions" in $[a,b]$ if it is not continuous in a finite number of points of this interval. Example: let $f(x) = \dfrac{1}{|x-b|^\alpha}$, with $\alpha < 0$ To retrieve $\displaystyle \int_a^b f(x) dx$, I must…
user3204810
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How to calculate $\int_2^4 \frac{\sqrt{\ln(9-x)}}{\sqrt{\ln(9-x)} + \sqrt{\ln(3+x)}} \space dx$

How to calculate $$\int_2^4 \frac{\sqrt{\ln(9-x)}}{\sqrt{\ln(9-x)} + \sqrt{\ln(3+x)}} \space\mathrm{d}x$$
user632241
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Value of integral involving $\tan x$

$$ \int_{0} ^ {\pi/4} \frac {\tan^2x}{1+x ^2} dx$$ I have tried using $x=\tan \theta$. But in my opinion i am finding that this might not be closed form. Please help.
maveric
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Integration on Two Independent Poisson Process

$$\int_{\theta =0}^T \left\{ \sum\limits_{u=0}^{S_1-1} \sum\limits_{m=0}^{S_1-u-1} p( m,\lambda_{0,1}(T-\theta)) p(u,\lambda_1\theta) \right\}\lambda _0 p(S_0-1,\lambda_0\theta ) \,d\theta $$ I am trying to work out this integral, I will appreciate…
Eln
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Definite integral of derivative of function

If I have the definite integral of the derivative of a function, is it the same as having the derivative of the definite integral of a function? For the latter, it would be the derivative of the upper bound*the inside function at that upper…
user592234
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Integration of some root functions.

while trying to help the daughter of my cousin to understand Integrals, I see that I forget about some of the integral calculations. Can someone help me with that one, please? $$\int_0^1 \Bigl(\sqrt[3]{(1-x^7)}-\sqrt[7]{(1-x^3)} \Bigr) dx$$ (this is…
tanaydin
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