# Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

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### Double Integral in Polar Form. Express $D$ in polar coordinates.

Let $D$ be the region bounded by $y=x^2$ and $y=2$. Express $D$ in polar coordinates. I would like to ask how to covert it to polar coordinates? How to make use of the two equations to know what is ...
• 133
1 vote
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### Evaluate: $\int_0^2\int_0^{2-x}(x+y)^2 e^{2y\over x+y}dy dx$.

Evaluate: $$\int_0^2\int_0^{2-x}(x+y)^2 e^{2y\over x+y}dy dx.$$ My attempt: I've changed the order and got this as $$\int_{y=0}^2\int_{x=0}^{2-y}(x+y)^2 e^{2y\over x+y}dy dx.$$ and then ...
• 710
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### Evaluating $\int_{0}^{\infty}x^{a}e^{-x^{3}}dx$ Using Spherical Coordinates

(Motivation) I made this improper integral for fun: $$\int_{0}^{\infty}x^{a}e^{-x^{3}}dx = \frac{1}{3}\Gamma\left(\frac{a+1}{3}\right)$$ where $\Re(a) > -1$. After struggling with various ...
• 2,854
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### I Understand how to use chain rule and the power rule and how to use the chain within the power rule. NOT sure of which rule to apply sometimes. [closed]

I seem to misunderstand when to use/apply the chain vs Power rule. I know chain is for composite functions but I need more than that textbook definition. I need some more detailed explanations and ...
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• 10.2k
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### Analytic continuation of $\int_{-\infty + i \cdot p}^{\infty + i \cdot p} \exp\left[ -w^{2} +\left|w-y- i\cdot p\right|-y^{2}\right]\operatorname{d}w$

I am dealing with the following integral defined in Eq. $(40)$. I am kind of sure that steps from Eqs. $(40)$ to $(41)$, $(41)$ to $(42)$ are both fine. I tested on Mathematica step from Eq. $(42)$ to ...
• 41
1 vote
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### $\alpha(b) - \alpha(a) \geq \int_a^b \alpha'(x) dx$ for $\alpha$ monotone and differentiable a.e.

This is a problem from Reed and Simon's book on functional analysis, where you are asked to prove $\alpha(b) - \alpha(a) \geq \int_a^b \alpha'(x) dx$ for a monotone function $\alpha$ that is ...
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• 2,909
1 vote
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### How to compute this integral (over a Lie group)?

I met the following question and I don't know how to compute it directly: Let $G:=GL(2,\mathbb{R})$, and equip $G$ with the Haar measure $$dg:=\frac{1}{|det(g)|^2}dg_{11}...dg_{22},~(g:=(g_{ij})).$$ ...
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### I know a function with finite number of discontinuity points is integrable, but this question is different [closed]

Let f be a bounded function on a bounded interval [a, b], and f is continuous on [a, b] except at a sequence of points {an : n = 1,2,···} ⊂ [a,b]. Assume limn→∞ an = c. Show that f is integrable over [...
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### Integral of $\int f^2(x)\,dx$
Can we write the integral $$\int f^2(x)\,dx$$ in terms of $f(x)$, $x$ and $\int f(x)\,dx$? I've tried solving the integral by parts, but I can only end up with an equation of the form $0=0$. Maybe it ...