# Questions tagged [substitution]

Questions that involve a replacement of variable(s) in an expression or a formula.

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### Show equality between these two series

I have a simple (?) index substitution (?) problem, where I want to show that $$\sum_{n=0}^\infty \sum_{m=0}^n a_n b_m c_{n-m} = \sum_{j=0}^\infty \sum_{k=0}^\infty a_{j+k} b_k c_j$$ but whatever I ...
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### Problems with integral substitution that involves weird physics index notation and distributions.

I‘m supposed to show that the 3x3 matrix T, whose components are given by: $$T^{ij}=\int_{\partial B_1(0)} \frac{x^i x^j}{|x||x|}d \Omega$$ ($x^i$ the i’th component of $x$, $\partial B_1(0)$ the ...
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### Difficulty understanding the substitution $dS(y)=rdS(z)$

In the proof for the mean value property for harmonic functions we start with the integral $\frac{1}{2\pi r}\int_{\partial B(x,r)}u(y)dS(y)$ where $\partial B(x,r)$ is the boundary of the disk $B(x,r)$...
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### Substitution in 2 variable limits

Goodmoring, I'm having difficulty in resolving 2 variable limits with some variable substitution. I can't understand when the substitution is legit or not. My calculus teacher told me that I've to ...
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### Convergence of improper integral, $\cos(1/x)$

I'm trying to deduce weather this improper integral is convergent or not: $$\int_{0}^{1}\dfrac{\cos(\frac{1}{x})}{x}dx.$$ I've tried using Dirichlet's test for convergence, yet I cant seem to ...
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### Effective method to solve $\frac{\sqrt[3]{1+x} -\sqrt[3]{1-x}}{\sqrt[3]{1+x} +\sqrt[3]{1-x}} = \frac{x(x^2+3)}{3x^2+1}$

I want to know, is there an easy method to solve below equation $$\frac{\sqrt[3]{1+x} -\sqrt[3]{1-x}}{\sqrt[3]{1+x} +\sqrt[3]{1-x}} = \frac{x(x^2+3)}{3x^2+1}$$ I tried it by plotting and find the ...
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### Integration by substitution: use of appropriate substitutions with roots of even index

I have the following integrals $$\int x\sqrt{x-1}\, dx, \quad \text{and} \quad \int \sqrt{e^x-1}\,dx$$ I am not interested in solving integrals but rather in a consideration that I have thought when I ...
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### What is the idea behind Subsitution? [closed]

What is the Logical formulation of Substitution? Specifically how do we treat looking at a formula and think about numerical values, or truth values. What is the idea behind our treatment of ...
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### Limits of integral over a region after substitution

I have the integral of a function $f(x,y,z)$ defined over the region $a<\sqrt{x^2-y^2-z^2}<b$; that is, $$I\equiv \int_{a<\sqrt{x^2-y^2-z^2}<b}dxdydzf(x,y,z).$$ I realized that defining ...
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### How to integrate $\frac{\cos2x}{ \sin x + \sin 3x}$?

I thought of using trig identities to get rid of $\cos2x$ and $\sin3x$, then use Weierstrass substitution, but I got myself into big trouble as the expression got too complicated. Is there a simpler ...
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### integral after substituion of $x\to-y$

I just cant spot the mistake in my calculation. As I said in the title, I used the substituion $x\rightarrow-y$, $dx\rightarrow-dy$ and $a<b$ \begin{align} F(b)-F(a)&=\big[F(x)\big]_a^b=\int_a^...
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### Integration by substitution, $\int_1^a f(x^s) dx = \int_1^{a^s} f(x) \dfrac{1}{sx^{1-1/s}} dx$

I am trying to show that, for a function $f: \mathbb{R} \to \mathbb{R}$, and any $s>0$, $a>1$, we have that $$\int_1^a f(x^s) dx = \int_1^{a^s} f(x) \dfrac{1}{sx^{1-1/s}} dx.$$ I am trying to ...
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### How can I make this 1st order ODE separable?

$$y (2+3xy) \,{\rm d} x = x (2-3xy) \,{\rm d} y$$ I tried using the substitution $y=\frac{v}{x}$, but that didn't get me far. Then I tried using the substitution $y=vx$, but that didn't work either. ...
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### How can I proceed from here?

$$\frac{dy}{dx}=\frac{2x+3}{y+x-2}$$ Using the substitution: $y=vx$ I turned this expression into $$\frac{dv}{dx}\cdot x +v=\frac{2x+3}{vx+x-2}$$ Usually, at this stage, you can cancel a few things ...
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### How do I know which function to substitute the variable when solving problems by the "Integration by substitution" method?

In Calculus, we use the "Integration by Substitution method" to integrate variables that are otherwise difficult to do by the conventional method. What I don't understand which function do ...
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### How does $u$-substitution work: $\int \frac{\sqrt y e^{-y/2}}{2} \mathrm{d}y$

I have been banging my head on the table for 3 hours trying to understand the final step here. Where did $-2^{(3/2)}$ come from? Why is there a $-u$ inside the square root?
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### Substituting $it+\ln t$ with $u$ in challenging integration procedure

Regarding this post I would like to ask the community for hints regarding this problem. Since the steps done on that original post, did not yield the correct result, I have done a different approach. ...
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How to solve this integral?$$\int\frac{\sec^2(x)}{(4+\tan^2(x))^2}\, dx$$ I've tried the following: Starting by substituting $\tan(x) = 2\tan(\theta)\implies \sec^2(x)\ dx= 2\sec^2(\theta)\ d\theta$ $$... • 1,810 4 votes 3 answers 152 views ### Why is this secant substitution allowed? On Paul's Math Notes covering Trig Substitutions for Integrals we start with an integral:$$\int{{\frac{{\sqrt {25{x^2} - 4} }}{x}\,dx}}$$Right away he says to substitute x=\frac{2}{5}\sec(θ). Why ... • 853 2 votes 2 answers 85 views ### is this definite integral correct? I have this definite integral$$ \int^{2}_{1} \frac{e^{1/x}}{x^4}dx$$this is my attempt: I used u-substitution. u = 1/x, and then -du = 1/x^2 dx I rewrote 1/x^4 as (1/x^2) * (1/x^2) now, I ... 1 vote 2 answers 55 views ### Validity of trigonometric substitutions Let us look at a term \dfrac{x}{\sqrt{1+x^2}}. Here x>0. Now we can make a trigonometric substitution x=\tan A. But why does this A have to be in (0,\frac{\pi}{2})? I don't understand ... 0 votes 1 answer 110 views ### Integrating \frac{0.36h^2 + 1.44h + 1.44}{0.034 - 0.012136 \sqrt{h}} - Stuck with one part I have been trying to follow how to integrate \frac{0.36h^2 + 1.44h + 1.44}{0.034 - 0.012136 \sqrt{h}} using https://www.integral-calculator.com/ , but when it gets to (2 \div 5295931061521 \times ... 2 votes 1 answer 58 views ### (Integration) Question on U-Substitution Mistake For context, I was playing with this integral recreationally:$$\int{20\sin(\frac{x^2}{35})}dx$$I decided to try u-substitution, and got the following:$$u=\frac{x^2}{35}\space,\space du=\frac{2x}{35}...
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Problem I need to compute the following interval \begin{equation*}\int_{t_\text{s}}^{t_\text{e}} \cos\left(a+b\tau+c\tau^2\right)\text{ d}\tau\end{equation*} where $t_{\text{s}},t_{\text{e}},a,b,c$ ...