Questions tagged [substitution]

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

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37 views

Is it possible to turn this 'proof' of the product rule into a rigorous argument?

I have often found linear approximation to be useful in understanding the main theorems of calculus. I tried using it to 'prove' the product rule, as I find the typical proof for it to be unintuitive. ...
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2answers
31 views

L'Hospital rule to find limit on indeterminate form.

I have the following limit to compute: $$\lim_{x\to 0}{\left(\cos x -1\over {5 x^2}\right)}$$ I need to find the limit as $x \to 0$. I tried using L'Hospital rule , so I found the derivative of the ...
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4answers
65 views

Solve $\int \frac{5}{4x^2+3}dx$

I've tried few way to resolve $\int \frac{5}{4x^2+3}dx$ but I think there's somthing I'm missing. As a first step I've took the constant out: $5\int \frac{1}{4x^2+3}dx$. Next I've thought it would be ...
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1answer
22 views

How to integrate with given substitution?

Using the substitution $u = x^2e^{-4x} + 3$, find $$\int\frac{x(1-2x)}{x^2+3e^{4x}} dx$$
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1answer
29 views

How does a polynomial transform upon making a substitution into it?

My title is very poorly worded, I apologise, I'm having a hard time wording the question. My lecturer described it as being obvious, so I must be missing something very fundamental (the substitution ...
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1answer
28 views

Trig substitution one-to-one

I know that when using integration by substitution, one needs to be careful to use one-to-one substitution. However, the following integration bothers me: $$\begin{align} \int_0^{2\pi}\sin^2\theta d\...
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1answer
35 views

Meanings of substituting variables in polynomials?

When I am substituting variables of a polynomial, does that have any special meaning? Are there things I can't do or need to know about - before doing such a thing, to, say, solve an equation, or for ...
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0answers
58 views

How to solve $\int e^{-\frac{\sqrt{x^2+z^2}-R}{H}}dx$?

How to solve $$ \int_a^b e^{-\frac{1}{H}(\sqrt{x^2+z^2}-R)}dx$$ z, R, H are constants, a and b are known. I tried the substitution $$\displaystyle u=-\frac{\sqrt{x^2+z^2}-R}{H}$$ $$\displaystyle du=-\...
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2answers
47 views

Why can I substitute x = a sin θ?

I am reading my Calculus book, Calculus 8th Edition by James Stewart, and in 7.3 (pp. 526), it explains that I can use the reverse substitution: $$x=a\sin(\theta)$$ for the integral: $$\int{\sqrt{a^{2}...
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2answers
54 views

How to arrive at $\sin\theta + c$ from $\int {\mathrm{dx}\over x\sqrt{x^2-2x}}$?

$$\int {\mathrm{dx}\over x\sqrt{x^2-2x}}$$ transform $x^2-2x$:$$\int {\mathrm{dx}\over x\sqrt{(x-1)^2-1}}$$ substituting x with u:$$\int {\mathrm{du}\over (u+1)\sqrt{u^2-1}}$$ substituting u with $\...
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1answer
55 views

tight asymptotic bound for $T(n) = 2T(n/4 - 100) + \sqrt n $

I just learned solving recurrence relation using substituion method. I am currently stucked in this question. I need to find a tight asymptotic bound for $$ T(n) = 2T(\frac{n}{4} - 100)+ \sqrt n$$ ...
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2answers
41 views

Show the integral : $\int_{0}^{\infty}\ln^2\Big(e^{-x}+1\Big)dx=\frac{\zeta(3)}{4}$ [duplicate]

I want to show that : $$\int_{0}^{\infty}\ln^2\Big(e^{-x}+1\Big)dx=\frac{\zeta(3)}{4}$$ I have tried substitution like $y=e^{-x}$ it gives : $$\int_{0}^{1}\frac{\ln^2(y+1)}{y}dy=\frac{\zeta(3)}{4}$$ ...
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4answers
79 views

Why can I use u-substitution in $\int \frac{x^2}{x^3-7}dx$?

Here is the problem from the textbook: $$\int \frac{x^2}{x^3-7}dx$$ I don't understand why u substitution works on this problem, as in the explanation from the textbook I can only use it when I have ...
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1answer
33 views

Finding Sum of Series with $U$-substitution

Find the sum of the series $$\sum_{n=1}^\infty \frac1{2^n} \int_1^2 \sin \left(\frac{\pi x}{2^n}\right) dx.$$ HINT: Simplify the $n$-th term of the series by making the substitution $u = x/2^n$ in the ...
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0answers
45 views

Why does substitution work in the case of derivatives?

I foraged this question on Google, but couldn't find any site explaining why the substitution works in the case of derivatives. On this site, there are questions regarding why it works in the case of ...
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4answers
46 views

Integrate $\int \frac {dv}{\frac {-c}{m}v^2 - g \sin \theta}$

I try to integrate $$\int \frac {dv}{\frac {-c}{m}v^2 - g \sin \theta}$$ I did substituted $u = \frac{c}{m}$ and $w = g \sin \theta$ to get $$-\int \frac {dv}{uv^2 + w}$$ I'm wondering if I have to do ...
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0answers
17 views

Integration by substitution of surjective function

I have a surjective function $g:\mathbb{R}\rightarrow S$ for some $S \subset\mathbb{R}$, a bijective function $f:\mathbb{R}\rightarrow\mathbb{R}$ and a constant known value $c$. I want to calculate ...
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1answer
20 views

In solving recurrence by substitution why do the inequality signs switch and what does it mean to not break inequality?

I have a recurrence $T(n) = 3T(\lfloor n/4 \rfloor) + cn^2$. The guess is that it is that $T(n) = O(n^2)$. The book then solves it like below for some constants $c > 0$ and $d > 0$ $$ T(n) \leq ...
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1answer
70 views

Integrate $\ln(\sqrt{1-x}+\sqrt{1+x})$

Integrate $$\ln(\sqrt{1-x}+\sqrt{1+x})$$ My attempts: i) Multiplying and dividing its conjugate, we get $$I=\int \ln(-2x)dx-\int\ln(\sqrt{1-x}-\sqrt{1+x})dx$$ The second term is equally challenging ...
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1answer
33 views

Substitution method for solving recurrence $T(n−1)+n $

I am learning substitution method for solving recurrence and there is the recurrence $T(n−1)+n$ and we need prove that it is $O(n^2)$ and we guess that $T(n) \leq cn^2$. So we solve for $$T(n) \leq c(...
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1answer
35 views

When to use polar substitution?

I have solved the problem below, which gave me this question: How do i know when to use polar substitution? If, instead of polar substitution, I directly set (x, y) = (0,0), I get division by zero. ...
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0answers
28 views

Semigroup of bit-string substitutions - any pointers?

Consider a pair $s=(s_0,s_1)$ of bit-strings (strings of 0s and 1s). Let $s$ act on a bit-string $b$ by replacing every $0$ in $b$ by $s_0$ and every $1$ in $b$ by $s_1$. Then the set $S$ of all such '...
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2answers
85 views

Why does $u$-substitution require the variable to be injective? [closed]

I read this answer but I didn't understand it. I expect a simple yet satisfying answer Why does $u$-substitution require the variable to be injective? What's the reason for that? I didn't understand.
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0answers
66 views

Solving non-linear PDE $(\tfrac{\partial t}{\partial x})^{2}+(\tfrac{\partial t}{\partial y})^{2}=f(y,t)$

A solution of the relatively complex problem led to the following non-linear PDE of the first order $$\left(\frac{\partial t}{\partial x}\right)^{2}+\left(\frac{\partial t}{\partial y}\right)^{2}=\...
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1answer
23 views

Integral with substitution and probability distribution function

I have the following integral: $$ \int_{0}^{n} (a x+b) g(x) dx $$ $g(x)$ is a probability density function. and $x= \epsilon + c$ where $c$ is deterministic and $\epsilon \sim f$. I want to rewrite ...
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1answer
59 views

What trick was used to transform this integral?

I found this transformation in the solution set for some integration exercises. It looks like substitution, but it's written so I don't understand what the substitution was. $$\int_0^1 \frac{-2x_2^2}{(...
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1answer
47 views

Integral substitution, convert integral [closed]

I need to find a good substitution to convert integral $\int \frac{1}{t^2+a^2}dt$ to integral $\int \frac{1}{x^2+1}dx$. Can anybody please help me? I don't know the method to use to I approach it. I ...
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1answer
50 views

Why does the boundary change on definite integrals substitution

I have this following Integral: $$\int_0^2 (2x-1)^3 dx$$ I want to integrate it using u-substitution, like that: $$u = 2x-1 $$ $${\frac{du}{dx}(2x + 1)} = 2$$ $$ {du = (2){dx}}$$ $$1/2\int (u)^3{du} $$...
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1answer
44 views

Show that when applying these substitution rules the result is an alternating sum with binomial coefficients in the numerators.

The following is Mathematica code. I apologize but I don't know how to write it more clearly. B1 = ReplaceAll[A1, x1 -> 1/A2]; simply means to take ...
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1answer
52 views

Substitution variables in Taylor series

I have troubles understanding why and when you can substitute your variables in a Taylor series. Could somebody help me explain why that is possible? Especially because the derivative often involves ...
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1answer
40 views

Change integration variable from scalar to matrix

Suppose $c$ is a scalar, $\mathbf{A}$ is a symmetric positive definite matrix and $g(.)$ is some real-valued function. Define $\mathbf{B} = c \, \mathbf{A}$. In this integral, $$ \int_{-\infty}^{\...
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1answer
46 views

Help with substitution

I am having troubles with the following transformation. I have: \begin{align*} Y_t^k=\int_0^t(t-s)^{\alpha-1}h(s)ds+\int_0^t (t-s)^{\alpha-1}\int_0^s(s-u)^{-\alpha}\sigma_{n_k}(X_u^{n_k})dB_u^{n_k}ds \...
1
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3answers
57 views

Evalution of a function where $t = x + \frac{1}{x}$ [duplicate]

Consider a function $$y=(x^3+\frac{1}{x^3})-6(x^2+\frac{1}{x^2})+3(x+\frac{1}{x})$$ defined for real $x>0$. Letting $t=x+\frac{1}{x}$ gives: $$y=t^3-6t^2+12$$ Here it holds that $$t=x+\frac{1}{x}\...
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1answer
20 views

Arc length of function - confused about particular u-substitution in integral

I am trying to teach myself how to compute the arc length of a function through a textbook, and I am stuck on how they did a particular integral substitution in a worked example. I'll provide the ...
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3answers
114 views

Integrating $\int \sqrt{a^2+x^2} \ \mathrm{d} x$ with trig. substitution

I am trying to come up with all the formulas I have myself and I stumbled upon a roadblock again. Integrating $\int \sqrt{a^2+x^2} \ \mathrm{d} x$ with Trig Substitution. So I imagined a triangle ...
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3answers
98 views

Why does making the 'wrong' $u$-substitution still work in this example?

I was evaluating the integral $$ \int 2x \cos(x^2)dx $$ and realised that it could be written in the form $$ \int f'(g(x))g'(x)dx $$ and so substitution could be used to help evaluate it. Setting $u$ ...
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3answers
72 views

Prove this tabulated integral $\int_0^\infty x^ne^{-\alpha x} \, dx=\frac{n!}{\alpha^{n+1}}$ [closed]

I ran into this problem where I needed to use the following integral equality in my physics textbook. $$\int_0^\infty x^ne^{-\alpha x} \, dx=\frac{n!}{\alpha^{n+1}}$$ where $n$ is a positive integer ...
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3answers
105 views

Prove that $(a b+b c+c a-1)^{2} \leq\left(a^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right)$.

Let $a$, $b$, and $c$ be real numbers. Prove that $$(a b+b c+c a-1)^{2} \leq\left(a^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right)\,.$$ In solution of this author take Let $a=\tan x, b=\tan y, ...
3
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0answers
87 views

Need some help with solving this integral…

I have to solve the following integral as part of another bigger expression: $$ \int_{A}^{1}{x \over 1 - x}\,(x - A)^{-\varepsilon} \ln(x)\left\{1+2\left[\ln(x) - {1 + x \over x}\,\ln(1 - x)\right]\...
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0answers
34 views

Different answers when changing variables in a definite integral with an infinite lower limit

I've searched for a post about this, to no avail. I am pretty sure I know which method is correct, but I'm having trouble pinning down exactly why the incorrect method is wrong. I wish to evaluate the ...
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0answers
45 views

Substitution inside integral?

Let $X$ be a random variable. I am trying to prove a special case of LOTUS (assuming $g$ is increasing and differentiable) using transformation of PDF. I have in my proof arrived at the following ...
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2answers
29 views

Expressing $\cos(\varphi)$ using $ z=e^{i \varphi} $

I need to express $ \cos(\varphi) $ with $z = e^{i \varphi}$ in order to use the Cauchy integral formula on the following integral: $ \int^{2 \pi}_0 \frac{1}{3+2\cos(\varphi)} \,d \varphi $ I got: $ \...
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1answer
39 views

Proof appears to use substitution inconsistently.

A textbook uses induction to prove the binomial theorem and uses the following substitution: $$\sum_{k=0}^{n-1} {n-1 \choose k} x^{k+1}y^{n-1-k} + \sum_{k=0}^{n-1} {n-1 \choose k} x^ky^{n-k}$$ Letting ...
4
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10answers
253 views

How can I integrate $\int \frac{u^3}{(u^2+1)^3}du?$

How to integrate following $$\int\frac{u^3}{(u^2+1)^3}du\,?$$ What I did is here: Used partial fractions $$\dfrac{u^3}{(u^2+1)^3}=\dfrac{Au+B}{(u^2+1)}+\dfrac{Cu+D}{(u^2+1)^2}+\dfrac{Au+B}{(u^2+1)^3}$$...
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4answers
74 views

$\int_{0}^{1}\frac{x}{\sqrt{1-x}}dx$

Solvable fixing $1-x=t$, I have a doubt about the integration's extremes. If $x=1\rightarrow t=1-x=0$, while if $x=0\rightarrow t=1-x=1$, so we have $-\int_{1}^{0}\frac{1-t}{\sqrt{t}}dt=\int_{0}^{1}\...
2
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2answers
86 views

Understanding the substitution theorem of Riemann integration.

Let us say $f$ is an integrable function on $[a,b]$ and we want to evaluate $\int_a^b f(x)dx$ but often the calculation is not easy.So,we have a method of substitution.We substitute $x=\phi(t)$ where $...
2
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2answers
183 views

How can I solve this $\int\dfrac{1}{\sqrt{5e^{-2x}+4e^{-x}+1} } \mathop{dx}=?$

How can I solve this $$\int\dfrac{1}{\sqrt{5e^{-2x}+4e^{-x}+1} } \mathop{dx}=?$$ My attempt: I substituted $e^{-x}=t$, $-e^{-x}\ dx=dt$, $dx=-\dfrac{dt}{t}$ $$\int\dfrac{1}{\sqrt{5t^2+4t+1 } }\left(-\...
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5answers
125 views

How do I evaluate $\int{\frac{x^3}{x^2-1}dx}$ using trigonometric substitution?

How do I use trigonometric substitution to rewrite and evaluate $$\int{\dfrac{x^3}{x^2-1}dx}?$$ I have no trouble using long division of $x^3$ by $x^2-1$ which is $x+\dfrac{x}{x^2-1}$ to get the ...
2
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1answer
26 views

Maximising the argument of $\sin and \cos$, given a linear relation between them

Here is a problem I have been struggling with for a while, If $$4\sin\theta \cos\phi+2\sin\theta+2\cos\phi+1=0$$ where $\theta,\phi\in[0,2\pi]$, find the largest possible value of $(\theta+\phi)$ I ...
0
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1answer
63 views

Is non-capture avoiding substitution permitted in the lambda calculus?

In the lambda calculus (typed and untyped) "capture-avoiding substitution" is often defined. But this doesn't rule out non-capture avoiding substitutions unless we require that all ...

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