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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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Evaluating $\int_0^1\left(\int_0^\infty\frac{x^2}{(x^3z^3+1)(x^3+1)}dx\right)dz$

I was trying to obtain some interesting identity and refresh my mathematics, as same manner and tricks showed in [1] for modified functions (my modified function that I write in the integrand and my modified Fact, see My attempt if you want/need to…
user243301
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Integrals convergence and divergence

let $f$ be a fucntion defined on $R^+$, let S= $\int_a^\infty f(x)$ with $a\ge0$. I need to know if there exists a theorem that states that if : $\lim_{x\rightarrow \infty}f(x) \rightarrow c$ where $c\neq 0$ hence the integral diverges? and if not,…
mandez
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prove equality - definite integral of definite integral

I need help to prove this expression. I can check that it is correct but can not prove it. Please help. $$\int_0^x \int_0^t f(v) dv dt = \int_0^x (x-t) f(t) dt $$
sasaborg
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Problem related to indefinite integral in definite integration

I tried to use binomial expansion for this problem, but it was in vain..I need the right approach. Can anyone guide me with the right idea? $$\int_1^{\infty}\frac{\,dx}{e^{x+1}+e^{3-x}}$$
Vamsi Spidy
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How does the discriminant of a quadratic equation give the definite integral?

$$\int^7_1(x-1)(x-7)dx$$ Wee solve it by the usual method $$\large[\frac{x^3}3]_1^7+[7x]^7_1-[4x^2]^7_1$$ $$\frac{343}{3}-\frac13+49-7-196+4=-36$$ But if we take a look at the determinant, it is ${64-28}=36$. So taking the negative of it, we have…
Aditya Agarwal
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Transform integral limits

Assume that the following relation holds \begin{align} p\int^1_{\frac{p}{a}} f(\theta)d\theta = \int^p_a f(\theta)d\theta \end{align} where $p$ is some scalar and $a\in[0,1)$. Is there some general rule which confirms the example at hand? Actually I…
clueless
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Let $R$ be the region in the first quadrant bounded by the $x$ and $y$ axis and the graphs of $f(x)=\frac{9}{25}x+b$ and $y=f^{-1}(x)$

Let $R$ be the region in the first quadrant bounded by the $x$ and $y$ axis and the graphs of $f(x)=\frac{9}{25}x+b$ and $y=f^{-1}(x)$.If the area of $R$ is 49,then the value of…
diya
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$\int_1^2 \operatorname{cosec}^2 4t\; dt $ Evaluate this definite integral

$$\int_1^2 \operatorname{cosec}^2 4 t\;dt$$ Been trying to evalute this definite integral, although I can't find what $\operatorname{cosec}^2$ is?
joe
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Definite Integration of A Non-even Non-odd Function Over A Symmetric Interval

We know that when a function $f(x)$ is even then: $\int^a_{-a} f(x) \, dx =2\int^a_0 f(x) \, dx$ and when $f(x)$ is odd , then: $\int^a_{-a} f(x) \, dx=0$ My question is: What can we say about $\int^a_{-a} f(x) \, dx$ if $f(x)$ is not even nor…
Fabian
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Finding the equivalent definite integrals

$\int x^{m-1}(1-x)^{n-1} dx $ (x=0 to x=1 ) I came across this question.Option A I could prove correct .But is any of the other three options correct? What should be the approach ?
user220382
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Area under bijective decreasing function

Let $ f:[2,4]\to[3,5]$ be a bijective decreasing function,then find the value of $\int_{2}^{4}f(t) dt-\int_{3}^{5}f^{-1}(t) dt.$ I am not sure whether $\int_{2}^{4}f(t) dt=\int_{3}^{5}f^{-1}(t) dt$ or not.As $f(t)$ and $f^{-1}(t)$ are inverse of…
Vinod Kumar Punia
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Question about relation between integral and summation in this case

I have $N(t)=N_{\alpha_{n}}(t)=\#\lbrace n:\alpha_{n}\leq t\rbrace$. Let $\{\alpha_{n}\}$ be a positive sequence, tending to infinity. Let $ \varphi (t) $ be a differentiable, positive, and non-increasing function on $[1;0]$, such that $ \varphi…
Nguyễn Thùy An
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Double Integrals: How to choose appropriate limits of integration?

I'm trying to find the double integral of dxdy over the area bounded by $y=\ln(x)$, $y=e+1-x$, and $y=0$. I've drawn it out and I tried making the limits of $x: \ln(x) \to e+1-x$ and those of $y: 0 \to 1$ but I'm not getting the correct answer.
Zoe Pesce
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Find the value of $\int_1^2\frac{x^2-1}{(x^2+1+3x)(x^2-x+1)}dx$

Find the value of $\int_1^2\frac{x^2-1}{(x^2+1+3x)(x^2-x+1)}dx$ I tried but it was failed. Can you help me!
Road Human
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check the result of $\int_{1}^{e} \frac{x^2-3x^3+1}{2x^3}$

I'm trying to solve this definite integral $$\int_{1}^{e} \frac{x^2-3x^3+1}{2x^3}$$ but the result do not coincide with the one on my book and even in wolframalpha the result is different but I do not see any error in it. Here my…
Christian Giupponi
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