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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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Let $ f$ be a function such that $|f(u)-f(v)|\leq|u-v|$ for all real $u$ and $v$ in an interval $[a,b]$

Let $ f$ be a function such that $|f(u)-f(v)|\leq|u-v|$ for all real $u$ and $v$ in an interval $[a,b]$.Then: $(i)$Prove that $f$ is continuous at each point of $[a,b]$. $(ii)$Assume that $f$ is integrable on $[a,b]$.Prove…
diya
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Given that $ U_n=[x(1-x)]^n$ and $n\geq2$,$V_n=\int_{0}^{1}e^xU_ndx$,prove that $V_n+2n(2n-1)V_{n-1}-n(n-1)V_{n-2}=0$

Given that $ U_n=[x(1-x)]^n$ and $n\geq2$,$V_n=\int_{0}^{1}e^xU_ndx$,prove that $V_n+2n(2n-1)V_{n-1}-n(n-1)V_{n-2}=0$ I tried to solve it by integration by parts,taking $U_n$ as first function and $e^x$ as second…
Brahmagupta
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Prove that $(1)\frac{1}{2}\leq\int_{0}^{2}\frac{dx}{2+x^2}\leq\frac{5}{6}$
$(2)2e^{-1/4}<\int_{0}^{2}e^{x^2-x}dx<2e^2$

Prove that $(1)\frac{1}{2}\leq\int_{0}^{2}\frac{dx}{2+x^2}\leq\frac{5}{6}$ $(2)2e^{-1/4}<\int_{0}^{2}e^{x^2-x}dx<2e^2$ I tried to prove it but my answer is not correct. For first part,As $0\leq x\leq2\Rightarrow 2\leq…
Brahmagupta
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Find the range of the function,$f(x)=\int_{-1}^{1}\frac{\sin x}{1-2t\cos x+t^2}dt$

Find the range of the function,$f(x)=\int_{-1}^{1}\frac{\sin x}{1-2t\cos x+t^2}dt$ I tried to solve it,i got range $\frac{\pi}{2}$ but the answer is ${\frac{-\pi}{2},\frac{\pi}{2}}$ $f(x)=\int_{-1}^{1}\frac{\sin x}{1-2t\cos…
Brahmagupta
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If $f(x)=x+\int_{0}^{1}[xy^2+x^2y]f(y)dy$ where $x$ and$y$ are independent variable.Find $f(x).$

If $f(x)=x+\int_{0}^{1}[xy^2+x^2y]f(y)dy$ where $x$ and$y$ are independent variable.Find $f(x).$ I tried to solve it. $f(x)=x+\int_{0}^{1}[xy^2+x^2y]f(y)dy$ $f(x)=x+x\int_{0}^{1}y^2f(y)dy+x^2\int_{0}^{1}yf(y)dy$ I applied integration by parts but…
Brahmagupta
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Let $\alpha,\beta$ be the distinct positive roots of the equation $\tan x=2x$,then find $\int_{0}^{1}\sin \alpha x \sin \beta x$dx

Let $\alpha,\beta$ be the distinct positive roots of the equation $\tan x=2x$,then find $\int_{0}^{1}\sin \alpha x \sin \beta x$dx,independent of $\alpha$ and $\beta$. My Attempt $\tan \alpha=2\alpha$ $\tan \beta=2\beta$ Adding the two equations,we…
user1442
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$U_n=\int_{0}^{1}x^n(2-x)^ndx,V_n=\int_{0}^{1}x^n(1-x)^ndx,n\in N$

For $U_n=\int_{0}^{1}x^n(2-x)^ndx,V_n=\int_{0}^{1}x^n(1-x)^ndx,n\in N$,which of the following statement is/are true $(A)\ U_n=2^nV_n\hspace{1cm}(B)\ U_n=2^{-n}V_n\hspace{1cm}(C)\ U_n=2^{2n}V_n\hspace{1cm}(D)\ U_n=2^{-2n}V_n$ $V_n $does not change…
mathspuzzle
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How to prove $\int_{1}^{100} \lfloor \arctan x \rfloor dx = 100 - \tan 1$?

The question was to find $$\int_{1}^{100} \lfloor \arctan x \rfloor dx $$. I hesitated because I learnt from illustrations in my book that when there is step up function, it is compulsory to break it at integral limits. Did that mean I've to break…
user142971
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$\int_{0}^{\frac{\sqrt{2}-1}{2}}\frac{dx}{(2x+1)\sqrt{x^2+x}}$

$\int_{0}^{\frac{\sqrt{2}-1}{2}}\frac{dx}{(2x+1)\sqrt{x^2+x}}$ This is in the form of $\frac{1}{linear\sqrt{quadratic}}$.I put $x=\frac{1}{t}$ $\int_{\frac{2}{\sqrt2-1}}^{\infty}\frac{dt}{(2+t)\sqrt{t+1}}$Then put $t+1=p^2$ From now,it got…
mathspuzzle
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Infinite summation of area of region

For $j=0,1,2,\ldots,n$. Let $S_j$ be the area of region bounded by the $x$-axis and the curve $ye^x=\sin x$ for $j\pi\leq x\leq(j+1)\pi$. The value of $\sum\limits_{j=0}^\infty S_j$ equals to (A)$ \dfrac{e^\pi(1+e^\pi)}{2(e^\pi-1)}$ (B)$…
Vinod Kumar Punia
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The value of the definite integral

The value of the definite integral $\displaystyle\int\limits_0^\infty \frac{\ln x}{x^2+4} \, dx$ is (A) $\dfrac{\pi \ln3}{2}$ (B) $\dfrac{\pi \ln2}{3}$ (C) $\dfrac{\pi \ln2}{4}$ (D) $\dfrac{\pi \ln4}{3}$ I tried using integration by…
Brahmagupta
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Solving a definite integral of exponential integral function combined with exponentials and rational functions

Everybody, I need to solve a definite integral of exponential integral function combined with exponentials and rational functions, which is presented as follows: $\int\limits_a^\infty {z\exp \left( { - \left( {b - c} \right)z}…
junxv penn
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Why is the infinite (definite) integral of x^3 , divergent and not 0?

Why is the integral from negative inf to positive inf of x^3 divergent? It's an odd function from -a to a, and because its odd, its symmetric about y=-x, So shouldnt all the areas cancel and get 0?
Nawar Ismail
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Proof of multiple integral equation

I am trying to prove the following equation: $$ \int_a^x\int_a^{t_1}\cdots\int_a^{t_{n-1}}\,dt_n\cdots\,dt_2\,dt_1=\frac{(x-a)^n}{n!}$$ I am not really sure where to begin. I would appreciate any help.
Remuo Rissan
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Hint for solving a definite integral $\int_{-a}^{a}\frac{xdy}{(x^{2}+y^{2})^{\frac{3}{2}}}$

Can anyone provide a hint for solving this definite integral: $\int_{-a}^{a}\frac{xdy}{(x^{2}+y^{2})^{\frac{3}{2}}}$
bpr3003
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