# Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

12,998 questions
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### How to determine the radius when utilizing the disc method

Lately, I have been struggling to figure out how to determine the radius used in the formula $V=\pi r^2$ when finding the volume of a solid revolving around an axis. From my experience with solving ...
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### Integral with log of absolute value of sine

Show that $$\int_{-\pi/3}^{\pi/3} \log \vert 8 \sin(t/2) (1 + \sin t)^2 \vert dt = 0.$$ WolframAlpha claims that this is true. I've tried manipulating the integrand a bunch and various trig ...
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### Suppose that the average value on all intervals $[a,b]$ is equal to $f((a+b)/2)$. Prove that $f''(x) = 0$ for all $x \in \mathbb{R}$

I understand that $f(x)$ must be linear with a first derivative equal to a constant. I'm just not sure how I can use the mean value property of integrals to show something about $f''(x)$. The hint on ...
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### Remove even elements of partition of integration set

Suppose I am integrating a continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ in a measurable set $A\subseteq I$, where $I$ is an interval: $$\int_{A}f(x)dx$$ Now suppose I partition the set $I$ in $N$ ...
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### More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$

In this post, the OP asks about the integral, $$I = \int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$$ I. User DavidH gave a beautiful (albeit long)...
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### Hard and weird Integration in numerator and denominator

$$\frac{\int_0^\pi x^3\ln(\sin(x))\ dx} {\int_0^\pi x^2\ln(\sqrt{2}(\sin(x))\ dx}$$ In this problem , I'm unable to understand how to start. I tried applying By parts but I couldn't solve it . ...
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### How to set up a line integral of a vector field over the borders of two circumferences

So...I have found a problem where I have to solve the integral of the vector field: $F(x,y)=(\sin(x)ln(x)+y^2) a_x + (\cos(y)e^y-x^2) a_y$ Along the borders of the region bounded by the ...
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### Is $\int_0^1 \frac{x^n}{\sqrt{1-x^4}}dx$ convergent?

$$\int_0^1 \frac{x^n}{\sqrt{1-x^4}}dx$$ Near $0$ the expression inside is convergent, that is easy. Near $1$ looks like it approaches infinity when $n \ge 0$ But according to the book when $n \ge -1$ ...
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### How can I solve this kind of integral? [on hold]

Evaluate $$\int_{-\infty}^{\infty}\frac{1}{z}e^{-i(Az^2 - Bz)}dz,$$ where A and B are some coefficients.
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### Prove $\ln \int_0^1 f(x)dx \geq \int_0^1 \ln f(x) dx$.

Let $f(x) \in C[0,1]$, and $f(x)>0$ over $[0,1]$. Prove $$\ln \int_0^1 f(x)dx \geq \int_0^1 \ln f(x) dx.$$ If we denote $$F(x):=\ln \int_0^x f(t){\rm d}t-\int_0^x \ln f(t){\rm d}t, ~~~x \in[0,1]$$ ...
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### Can $1/\log(2)$ be represented as a period?

In this article by Zagier-Kontsevich, period is defined as values of integral of a rational function over a domain in $\mathbb{R}^{n}$ defined by polynomial inequalities with rational coefficients. ...
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### $\lim_{n \to \infty} n \int_{0}^{100}f(x)g(nx)dx=f(0)$

Let $g: \mathbb{R} \to \mathbb{R}$ be a continuous function with $g(y)=0$ for all $y \notin [0,1]$ and $\int_{0}^{1}g(y)dy=1$. Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function. ...
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### Can a certain series of integrals over $[0,\frac{1}{16}]$ be solved using integration-by-parts?

I have a series of six integrals. The first, say, is \begin{equation} \int_ 0^{\frac {1} {16}} - 48 q^{3/2}\sqrt {18 q - 2\sqrt {17 q + 4}\sqrt {q} + 4} \mbox {d} q. \end{equation} Mathematica ...
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### Solving double integral [closed]

Can someone help me to solve the following double integral; where $a_1,\epsilon_1 ,\Phi, p, a_2 ,k$ are constants and $a_2 -a_1 = \epsilon_1$; integral to solve
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### Solve the numerical value of this integral $\int_0^\infty t^{a-1}e^{-t} \Gamma(b,t)dt$

I need to compute the numerical value of this integral, hundred thousand of times, for a typical dataset. How can I get a good approximation. $$\int_0^\infty t^{a-1}e^{-t} \Gamma(b,t)dt$$ where a ...