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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.

20559 questions
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Definite integral of product of functions

I know it's not correct to write: $$\int_{a}^{b}f(x)g(x) dx = \int_{a}^{b}f(x)dx\int_{a}^{b}g(x)dx$$ This result seems obvious, but I can't think of a way to prove that $\int_{a}^{b}f(x)g(x) dx$ can't be expressed as a function of the form…
andynash
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Existence of the integral $\int_{0}^{\pi}\frac{\sin (x+\sqrt x)}{\cos(x-\sqrt x)} dx$

Can anyone help me in evaluating the following integral : $\int_{0}^{\pi}\frac{\sin (x+\sqrt x)}{\cos(x-\sqrt x)} dx$. I tried doing this by substitution but didn't work. Also by parts looks cumbersome.
creative
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Does the integral $\int_{a}^{b}\frac{dx}{\sqrt{(x-a)(x-b)}}$ exist?

What is the result of this integral $\displaystyle\int_{a}^{b}\dfrac{dx}{\sqrt{(x-a)(x-b)}}$ ? I have tried many possibilities like letting $\sqrt{(x-a)(x-b)}$=u or trying to make the denominator express as a difference of two sqares but nothing…
creative
  • 3,539
5
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Two definite integrals

These integrals are closely related since $\frac{\pi^2}{8}=\sum_{n=0}^{\infty} \frac{1}{(2n+1)^2}$ and $G=\sum_{n\mathop=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2}$. I'm not able to prove them though. Show that $$\int_0^1…
user85798
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Closed form for $\int_0^\alpha \int_0^{\alpha-x} \frac{1}{[(x+y)(\beta-x)(1-\beta-y)]^{3/2}} \,dy \,dx$

Let $0\leq \alpha \leq \frac{1}{2},$ and let $\beta$ be such that $\alpha \leq \beta \leq 1-\alpha$. Define $\varphi(\alpha,\beta)$ to be the integral $$\varphi(\alpha,\beta) = \int_0^\alpha \int_0^{\alpha-x} \frac 1…
Ted Dokos
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Change of integration order of a double integral

Which is the integral equivalent to $$\int_{0}^{\pi}\int_{0}^{\sin x}f(x, y)dydx$$ to make a change in the order of integration?
Roiner Segura Cubero
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Evaluate the definite integral $\int _0^1x^bb^x\,dx$.

Since, the question asks to evaluate $\displaystyle \int \limits _0^1x^bb^x\,dx$. Sharing my thought shots on the same: I multiplied and divided my integrand with $\log b^x$ and afterwards substituted $b^x=t$. So my Integrand reduced to…
Aastha Choudhary
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Definite integral: exponential times non-integer powers

I would like to calculate: $$ \int_0^1 \sqrt{1+ a x^2} e^{-a x^2/c} \, \mathrm{d}x $$ for any positive constants $a,c>0$. I am happy with any closed expression involving any kind of special functions. Failed attempts so far: standard changes of…
AlephBeth
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Find $\int_{0}^{1} f(x)^3\:dx$

If $f:[0,1]\to R$ satisfies: $$\int_{0}^{1}f(x)dx=1$$ $$\int_{0}^{1}xf(x)dx=1$$ $$\int_{0}^{1}f(x)^2dx=4$$ Find: $$\int_{0}^{1}f(x)^3dx$$ My try: Let $f(1)=p$ I used Parts: we get $$p-\int_{0}^{1}xf'(x)dx=1$$ $$p-\int_{0}^{1}(xf'(x)+f(x))xdx=1$$…
Ekaveera Gouribhatla
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Calculate $\int_0^1 \frac{-\ln(1-x)}{x} d x$ without $\zeta(2)$.

Can we calculate the following integral without the need of $\zeta(2)$, I actually believe that this can be a method to find the accurate value of $\sum_{n\geq 1}n^{-2}$. $$\int_0^1 -\frac{\log (1-x)}{x} \ \mathrm{d}x$$ Any help is appreciated ,…
Tulip
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Help evaluating integral (anything simple that I am missing?)

This integral is [hopefully] between 0 and 1 as it is supposed to represent a probability. $$\int_{-\infty}^{\infty} \int_{-2-x}^{2-x} \frac{1}{2\pi} e^{\frac{-x^2-y^2}{2}} dy dx$$ I just wanted to check if anyone saw any easy-ish method to evaluate…
user694996
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How to calculate the integral $\int\limits_{-10}^{10}\frac{3^x}{3^{\lfloor x\rfloor}}dx$?

I have to calculate this integral: $$\int\limits_{-10}^{10}\frac{3^x}{3^{\lfloor x\rfloor}}dx$$ I know that this function ie. $$3^{x-\lfloor x\rfloor}$$ is periodic with period $T=1$ so I rewrote the integral as $$20\int_{0}^{1}\frac{3^x}{3^{\lfloor…
infinite-blank-
  • 1,036
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How to evaluate $ \int_{0}^2 x^{26} (x-1)^{17} (5x-3)dx ~~ ?$

How to evaluate $$ \int_{0}^2 x^{26} (x-1)^{17} (5x-3)dx ~~ ?$$ I have tried to evaluate, $$ I = \int_{0}^2 x^{26} (x-1)^{17} (5x-3)dx = 5\int_{0}^2 x^{27} (x-1)^{17} dx - 3\int_{0}^2 x^{26} (x-1)^{17} dx$$ by parts but I am getting very lengthy…
Pallavi
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The integral $\int_{0}^{\pi/2} \sin 2\theta ~ \mbox{erf}(\sin \theta)~ \mbox{erf}(\cos \theta)~d\theta=e^{-1}$

In a work it was required to find an integral akin to $$I=\int_{0}^{\pi/2} \sin 2 \theta~\tanh(\sin\theta) \tanh(\cos \theta)~d\theta.$$ Since $\tanh x$ and $\operatorname{erf}(x)$ are similar functions so it was a pleasure to see that its variant…
Z Ahmed
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Find all values of $r$ if $\int_{0}^{\infty} \frac{dx}{(1+x^r)^r} =1. $

Find all values of $r$ if $$\int_{0}^{\infty} \frac{dx}{(1+x^r)^r} =1. $$ I have found one value of $r$ by a brute force method. I use the substitution $x^r=\tan^2 t$ to convert the required integral as: $$J=\int_{0}^{\infty} \frac{dx}{(1+x^r)^r}=…
Pallavi
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