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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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How to evaluate the definite integral $\int_0^\infty t^{n-1}e^{-at}dt$

How to evaluate the following definite integral $\int_0^\infty t^{n-1}e^{-at}dt$.
Litun
  • 670
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2 answers

integrals proving equality

Prove that $$\int_a^b f(x)\,dx = c\int_{a/c}^{b/c} f(cy)\,dy$$ I've tried to use subtitution rule, but I'm afraid I can't do this if f is not continuous. Thanks.
Shirly Geffen
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Differential equation and substitution.

If the substitution square root of $x = \sin y$ is made in the $\int^0_{0.5}\dfrac{\sqrt{x}}{\sqrt{1-x}}dx$, what is the resulting integral?
Hannah
  • 67
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How to integrate $\int^{\pi/2}_0\sin^4xdx$

Sorry if the question is lame but here it The following was given in my textbook $$\int^{\pi/2}_0\sin^4xdx$$ so i integrated it this way $$\implies\int^{\pi/2}_0\sin^4xdx = \int^{\pi/2}_0\frac{\sin^5x}{5}(-cosx)$$$ and then substituted the values…
Deiknymi
  • 383
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How is the following related to the Gamma function?

There's a shortcut formula in my book: $$ \int_{0}^{\pi/2}\sin^{m}\left(x\right)\cos^{n}\left(x\right)\,{\rm d}x = {\left[\left(m - 1\right)\left(m - 3\right)\ldots\,2\ \mbox{or}\ 1\right] \left[\left(n - 1\right)\left(n - 3\right)\ldots\,2\…
Shubham
  • 880
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Convex function in integral

Let $f:[0,\infty)\rightarrow [0,\infty) $ be a concave function and $ p\in [0,\dfrac{1}{2}]$ a) Show that f is integrable on each interval $[a,b] \subset [0,\infty).$ b) Show that $ F:[0,\infty)\rightarrow \Bbb{R}$, $F(x)=\int_{0}^{x^p} f(t)dt$ is…
Maish Petronas
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Definite Integral of function

I have this integral: $$\int_0^\infty \sqrt{x} e^{-\frac{x}{a}}dx$$ I am not sure how to solve this. I think it may involve the erf function (courtesy of wolfram) but I am not sure how to appropriately use this. The answer should…
user92089
  • 21
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Valid integration method? $\int_{-1}^{1}\frac{dx}{\sqrt{1-x^{2}}}=-i\left(\cosh^{-1}\left(1\right)-\operatorname{arcosh}\left(-1\right)\right)$

I've cherrypicked the appropriate branch of arcosh to allow the method to 'work'. Is this a valid integration method? Using the substitution $x=\cosh\theta$ we…
Simon M
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a Legendre integral identity

I was given the following problem. For $n\ge2$ natural and $z\in\mathbf{C}\setminus[-1,1]$, prove the Legendre identity $$\int_{-1}^{+1}\frac{(1-t^2)^{(n-3)/2}}{(z-t)^n}dt =…
Yul Otani
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Evaluation of $ \lim_{n\rightarrow \infty}\bigg(((n+2)!)^{\frac{1}{n+2}}-((n!))^{\frac{1}{n}}\bigg)$ using Definite integration

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\bigg(((n+2)!)^{\frac{1}{n+2}}-((n!))^{\frac{1}{n}}\bigg)$ using Definite integration My Try : Let we assume $\displaystyle l_1=\lim_{n\rightarrow…
jacky
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Number of real solution of expression containing definite integral

Let $f:[a,b]\rightarrow \mathbb{R}$ be a continuous function and $$\displaystyle \int^{b}_af(x)\sin\left(\frac{x-a}{b-a}\pi\right)\,dx=\int^b_af(x)\cos\left(\frac{x-a}{b-a}\pi\right)\,dx=0.$$ Then number of solution of $f(x)=0$ in $(a,b)$ is Let…
Priti Bisht
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The order of double integration of the $\int_0^\infty \frac{2a^2}{x^2 + a^2} (\int_0^1 \cos(yx) dy)dx$.

I have integral $\int_0^\infty \frac{2a^2}{x^2 + a^2} \frac{\sin x}{x}dx$ after the next transformation $\frac{\sin x}{x} = \frac{1}{x} \int_0^1 x\cos(yx) dy = \int_0^1 \cos(yx) dy$ I get a double integral $\int_0^\infty \frac{2a^2}{x^2 + a^2}…
Varga
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How to prove that $ \int_0^1 \sqrt{\ln(\frac{1}{x})} \ \frac{\vartheta _3(0,x)-1}{x} dx = \sqrt{\pi}\zeta(3) $?

Where $ \vartheta _3(0,x) $ is the elliptic theta function. I first tried to expand the series since the integrand is within the radius of convergence of the series to no avail. Also, I'm not sure an exponential substitution would make the Theta…
DecarbonatedOdes
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integral $\int_0^a \frac{1}{\sqrt{1+x^6}} dx$ where $a=\frac{1}{\sqrt{\sqrt{3}-1}}$

$I=\int_0^a \frac{1}{\sqrt{1+x^6}} dx$ where $a=\frac{1}{\sqrt{\sqrt{3}-1}}$ first, do binomial series on the integrand function, $\sum_{n=0}^\infty \binom{-\frac{1}{2}}{n}x^{6n}$ after integration, we got $\sum_{n=0}^\infty \frac{(-1)^n…
MathFail
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Maxima of $\int_0^a f(x)^2 \ dx$ and $\int_0^a xf(x)^2 \ dx$

Let $f:[0,1]\rightarrow[0,\infty)$ be an increasing function, $a \in (0,1)$, and $\displaystyle \int_0^1 f(x) \ dx =1 $. What are the maxima of $$i)\int_0^a f(x)^2 \ dx$$ $$ii)\int_0^a xf(x)^2 \ dx$$ Some clues? EDIT: maxima need to be expressed as…
Damn_crazy_girl
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