# Questions tagged [change-of-variable]

This concern all problem requesting techniques and tricks about changes of variables in both computation of limits and integrals

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### Why is $\int u\frac{dv}{dx} \, dx = \int u\ dv$? (Change of Variable while deriving Integration by Parts)

I have been learning the integration by parts formula. $$\int u\ \mathrm dv = uv \ - \ \int v\ \mathrm du$$ I understand how the formula is derived when we keep everything in terms of $x$ (with $f(x)$...
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### Equivalence with the Weierstrass transform

I have the following expression $$\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{+\infty}dx~ f(x-y) e^{-x^2/4 t} \tag{1},~~\forall ~y \in \mathbb{R}.$$. I am trying to relate it with the generalized ...
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### Most general conditions for variable substitution with Riemann integral

This question is motivated by the discussion here: https://matheducators.stackexchange.com/a/26687/117 Let $g$ be defined and differentiable on an interval containing $[a,b]$ and $f$ be defined on an ...
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### Finding the area bounded by some curves using change of variables and double integrals

find the area of the first-quadrant region bounded by the curves $y = x^3,y=2x^3,x=y^3,x=4y^3$ be the curves that bound the area: now we can use the substitution: $$y =ux^3$$ $$x =vy^3$$ in order to ...
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### How to understand change of variables intuitively?

I've been trying to prove or have an intuitive understanding of the change of variables, and I tried it for the function $f(x)=x^2$ using $u(x)=x^2$, the transformed function then becomes $g(u)=u$. ...
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### Finding double integral over balls

I am facing trouble in evaluating some double integral. The definition of Riesz energy is given by $I_t(U)=\int_U\int_U|x-y|^{-t}\ dx\ dy$ where $U$ is an open subset. For better understanding I want ...
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### Variable changes in matrix integrals

I need to evaluate the following matrix integral: $I = \int \mathrm{etr}\left(-cX^TX\right)\mathrm{det}(I_d + AX^TBX)^{-k/2}dX$ where $A$, $B$, and $X$ are $d\times d$, $A$ and $B$ are positive ...
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### Help with a variable change

I'm integrating a function of the variables $x_i,y_i,z_i$ for $i=1,2,3$ and I define new variables given by $$u_1=x_1+y_3 z_3,$$ $$u_2=x_2+y_1 z_1,$$ $$u_3=x_3+y_2 z_2.$$ NOTE: I don't actually want ...
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### joint density of a random variable and its bijective, differentiable function

Consider a random variable $X$ and its density function $f_{X}(x)$, consider a bijective, differentiable function $H$, and let the random variable $Y=H(X)$. I am trying to compute the joint density ...