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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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Evaluation of $\int^{\infty}_{0}\frac{x^3}{e^x-1}dx$

Evaluation of $$\int^{\infty}_{0}\frac{x^3}{e^x-1}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int^{\infty}_{0}\frac{x^3}{e^x-1}dx$ Now put $\displaystyle e^x=\frac{1}{t}\;,$ Then $\displaystyle e^xdx = -\frac{1}{t^2}dt$ and $x=-\ln t$ and…
juantheron
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How to show that $\int_0^x \left(\frac{1}{(t-1)^{n+1}}-\frac{1}{(t+1)^{n+1}}\right)(x-t)^n\mathrm dt\to0$ when $n\to\infty$, for $\vert x\vert < 1$?

I want to show that : $$ \forall \vert x\vert < 1 \:,\: \int_0^x \left(\dfrac{1}{(t-1)^{n+1}}-\dfrac{1}{(t+1)^{n+1}}\right)(x-t)^n\mathrm dt \xrightarrow[n\to\infty]{} 0 $$ I've tried to bound from above $(x-t) ^n$ but…
SuperFoxy
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Definite integrals-is the question wrong or my method?

This problem has been troubling me for a while now; even though I have tried my best ( and filled up a rough notebook in the process). Consider $I_1=\int_{0}^{1} \frac{\tan^{-1}x}{x} dx$, and $I_2=\int_{0}^{\pi/2} \frac{x}{\sin x}dx$. We are…
GRrocks
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Evaluation of Trigonometric Integral

Evaluation of $\displaystyle \int^{\frac{\pi}{4}}_{0}\frac{\sin^2 x\cdot \cos^2 x}{\sin^3 x+\cos^3 x}dx$ $\bf{My\; Try::} $ Let $$I= \int^{\frac{\pi}{4}}_{0}\frac{\sin^2 x\cdot \cos^2 x}{\sin^3 x+\cos^3 x}dx =…
juantheron
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A family of integrals equals to $\pi$

Let $P_{n,k}$ be the $k$-th number of the Pascal's triangle on the $n$-th row, that is $P_{n,k} = \binom {n-1}{k-1}$ For example, $(P_{3,1},P_{3,2},P_{3,3})=(1,2,1)$. I have noticed through random computations that the following integrals seems to…
E. Joseph
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Prove $\int_0^{\pi/2}\frac x {\sin x} \, \mathrm d x = 2\sum_{n\mathop = 0}^{\infty} \frac {(-1)^n}{(2n+1)^2}$

I would like to prove that $$\int_0^{\pi/2}\frac x {\sin x} \, \mathrm d x = 2\sum_{n\mathop = 0}^{\infty} \frac {(-1)^n}{(2n+1)^2}$$ Any hints?
user85798
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How to calculate $\int_{0}^1 \sqrt{\frac{1-x^2}{1+x^2}}dx$?

I encounter the following question: $$\int_{0}^1 \sqrt{\frac{1-x^2}{1+x^2}}dx=?$$ It looks simple. But I doubt there exists analytic solution. Could anyone help? Thank you! Icarus 369 has pointed out Page [ Evaluating…
kaiwu
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Integration Substitution Problem 2

Let us assume a definite integral: say, $\int_0^{\frac{\pi}{2}} \frac{dx}{5+4 \sin x}$. Normally by substitution of $\tan\frac{\theta}{2}$ as t I can prove the answer is $\frac{1}{3} \ln 2$. But suppose I do not know any mathematics and I am foolish…
SN77
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Ratio of $2$ definite Integrals

If $\displaystyle I = \int_{0}^{\pi}\frac{\sin (884 x)\sin (1122x)}{2\sin x}dx$ and $\displaystyle J = \int_{0}^{1}\frac{x^{238}(x^{1768}-1)}{x^2-1}dx\;,$ Then $\displaystyle \frac{I}{J}$ $\bf{My\; Try::}$ $$\displaystyle…
juantheron
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lemmas on integrals

I tried to prove the following lemmas: if $f,g$ are two bound functions in $[a,b]$, then: $\underline{\int_{a}^{b}}{f(x)dx}+\underline{\int_{a}^{b}}{g(x)dx}\leq\underline{\int_{a}^{b}}{(f+g)(x)dx}$ and let $c$ be a point in $(a,b)$,…
nono
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Integrate from zero to infinity 1/(xe^x)

I cannot solve the integral $$\int_{x=0}^{\infty}\frac{dx}{xe^x}.$$ I tried it by use integration by parts and gama function.
Safikamal Sk
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Relating Definite Integrals to Infinite Series

I am trying to prove results (a) and (b) for definite integrals, and have to carefully justify my steps. My question is NOT how to show these results can be obtained in other ways. The question is how do I justify switching the infinite sum and…
WoodWorker
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Definite integral $\int_{\frac{1}{a}}^a \frac{\arctan(x)}{x}$

I have to fund the value of the above integral $$\int_{\frac{1}{a}}^a \frac{\arctan(x)}{x}$$ for $a=2014$. I just saw that integral of $\frac{\arctan(x)}{x}$ does not have a closed form. So I used the substitution $x=tan(\theta)$ and then used…
Sayan Chattopadhyay
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$\sin ix$ integral identity

If you plug in $\sin ix$ to wolframalpha, you get this really weird integral identity back: $$\sin(ix) = \frac{x}{4\sqrt{\pi}}\int_{-i\infty+\gamma}^{i\infty+\gamma}\frac{e^{s+\frac{x^2}{4s}}}{s^{3/2}}ds, \ \gamma > 0$$ There is a whole package of…
Benjamin Lindqvist
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Symmetrizing an integral representation of a symmetric function

Here is a function of two complex numbers $a$, $b$ that I believe is symmetric under the exchange $a \leftrightarrow b$: $$ I(a,b) = (-a+1/b)\int_0^\infty dx \frac{\ln(1+bx)}{(x+a)(x+1/b)} $$ This integral representation does not exhibit this…
QuantumDot
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