Questions tagged [substitution]
Questions that involve a replacement of variable(s) in an expression or a formula.
1,513
questions
1
vote
2answers
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. ...
2
votes
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 ...
0
votes
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 ...
-1
votes
1answer
22 views
How to integrate with given substitution?
Using the substitution $u = x^2e^{-4x} + 3$, find $$\int\frac{x(1-2x)}{x^2+3e^{4x}} dx$$
2
votes
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 ...
0
votes
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:
$$\begin{align}
\int_0^{2\pi}\sin^2\theta d\...
0
votes
1answer
35 views
Meanings of substituting variables in polynomials?
When I am substituting variables of a polynomial, does that have any special meaning? Are there things I can't do or need to know about - before doing such a thing, to, say, solve an equation, or for ...
1
vote
0answers
58 views
How to solve $\int e^{-\frac{\sqrt{x^2+z^2}-R}{H}}dx$?
How to solve
$$ \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=-\...
2
votes
2answers
47 views
Why can I substitute x = a sin Īø?
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: $$\int{\sqrt{a^{2}...
0
votes
2answers
54 views
How to arrive at $\sin\theta + c$ from $\int {\mathrm{dx}\over x\sqrt{x^2-2x}}$?
$$\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}}$$
substituting u with $\...
0
votes
1answer
55 views
tight asymptotic bound for $T(n) = 2T(n/4 - 100) + \sqrt n $
I just learned solving recurrence relation using substituion method.
I am currently stucked in this question.
I need to find a tight asymptotic bound for $$ T(n) = 2T(\frac{n}{4} - 100)+ \sqrt n$$
...
0
votes
2answers
41 views
Show the integral : $\int_{0}^{\infty}\ln^2\Big(e^{-x}+1\Big)dx=\frac{\zeta(3)}{4}$ [duplicate]
I want to show that :
$$\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}$$
...
2
votes
4answers
79 views
Why can I use u-substitution in $\int \frac{x^2}{x^3-7}dx$?
Here is the problem from the textbook:
$$\int \frac{x^2}{x^3-7}dx$$
I don't understand why u substitution works on this problem, as in the explanation from the textbook I can only use it when I have ...
0
votes
1answer
33 views
Finding Sum of Series with $U$-substitution
Find the sum of the series $$\sum_{n=1}^\infty \frac1{2^n} \int_1^2 \sin \left(\frac{\pi x}{2^n}\right) dx.$$
HINT: Simplify the $n$-th term of the series by making the substitution $u = x/2^n$ in the ...
1
vote
0answers
45 views
Why does substitution work in the case of derivatives?
I foraged this question on Google, but couldn't find any site explaining why the substitution works in the case of derivatives. On this site, there are questions regarding why it works in the case of ...
1
vote
4answers
46 views
Integrate $\int \frac {dv}{\frac {-c}{m}v^2 - g \sin \theta}$
I try to integrate
$$\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}$$
I'm wondering if I have to do ...
0
votes
0answers
17 views
Integration by substitution of surjective function
I have a surjective function $g:\mathbb{R}\rightarrow S$ for some $S \subset\mathbb{R}$, a bijective function $f:\mathbb{R}\rightarrow\mathbb{R}$ and a constant known value $c$.
I want to calculate ...
0
votes
1answer
20 views
In solving recurrence by substitution why do the inequality signs switch and what does it mean to not break inequality?
I have a recurrence $T(n) = 3T(\lfloor n/4 \rfloor) + cn^2$. The guess is that it is that $T(n) = O(n^2)$. The book then solves it like below for some constants $c > 0$ and $d > 0$
$$ T(n) \leq ...
1
vote
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 ...
0
votes
1answer
33 views
Substitution method for solving recurrence $T(nā1)+n $
I am learning substitution method for solving recurrence and there is the recurrence $T(nā1)+n$ and we need prove that it is $O(n^2)$ and we guess that $T(n) \leq cn^2$.
So we solve for
$$T(n) \leq c(...
1
vote
1answer
35 views
When to use polar substitution?
I have solved the problem below, which gave me this question: How do i know when to use polar substitution?
If, instead of polar substitution, I directly set (x, y) = (0,0), I get division by zero. ...
1
vote
0answers
28 views
Semigroup of bit-string substitutions - any pointers?
Consider a pair $s=(s_0,s_1)$ of bit-strings (strings of 0s and 1s). Let $s$ act on a bit-string $b$ by replacing every $0$ in $b$ by $s_0$ and every $1$ in $b$ by $s_1$. Then the set $S$ of all such '...
0
votes
2answers
85 views
Why does $u$-substitution require the variable to be injective? [closed]
I read this answer but I didn't understand it. I expect a simple yet satisfying answer
Why does $u$-substitution require the variable to be injective? What's the reason for that? I didn't understand.
2
votes
0answers
66 views
Solving non-linear PDE $(\tfrac{\partial t}{\partial x})^{2}+(\tfrac{\partial t}{\partial y})^{2}=f(y,t)$
A solution of the relatively complex problem led to the following
non-linear PDE of the first order
$$\left(\frac{\partial t}{\partial x}\right)^{2}+\left(\frac{\partial t}{\partial y}\right)^{2}=\...
0
votes
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 ...
0
votes
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.
$$\int_0^1 \frac{-2x_2^2}{(...
-4
votes
1answer
47 views
Integral substitution, convert integral [closed]
I need to find a good substitution to convert integral $\int \frac{1}{t^2+a^2}dt$ to integral $\int \frac{1}{x^2+1}dx$.
Can anybody please help me? I don't know the method to use to I approach it.
I ...
0
votes
1answer
50 views
Why does the boundary change on definite integrals substitution
I have this following Integral:
$$\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} $$...
0
votes
1answer
44 views
Show that when applying these substitution rules the result is an alternating sum with binomial coefficients in the numerators.
The following is Mathematica code. I apologize but I don't know how to write it more clearly.
B1 = ReplaceAll[A1, x1 -> 1/A2]; simply means to take ...
2
votes
1answer
52 views
Substitution variables in Taylor series
I have troubles understanding why and when you can substitute your variables in a Taylor series. Could somebody help me explain why that is possible? Especially because the derivative often involves ...
3
votes
1answer
40 views
Change integration variable from scalar to matrix
Suppose $c$ is a scalar, $\mathbf{A}$ is a symmetric positive definite matrix and $g(.)$ is some real-valued function. Define $\mathbf{B} = c \, \mathbf{A}$.
In this integral,
$$
\int_{-\infty}^{\...
1
vote
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
vote
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 $$t=x+\frac{1}{x}\...
0
votes
1answer
20 views
Arc length of function - confused about particular u-substitution in integral
I am trying to teach myself how to compute the arc length of a function through a textbook, and I am stuck on how they did a particular integral substitution in a worked example.
I'll provide the ...
1
vote
3answers
114 views
Integrating $\int \sqrt{a^2+x^2} \ \mathrm{d} x$ with trig. substitution
I am trying to come up with all the formulas I have myself and I stumbled upon a roadblock again.
Integrating $\int \sqrt{a^2+x^2} \ \mathrm{d} x$ with Trig Substitution.
So I imagined a triangle ...
1
vote
3answers
98 views
Why does making the 'wrong' $u$-substitution still work in this example?
I was evaluating the integral
$$
\int 2x \cos(x^2)dx
$$
and realised that it could be written in the form
$$
\int f'(g(x))g'(x)dx
$$
and so substitution could be used to help evaluate it. Setting $u$ ...
1
vote
3answers
72 views
Prove this tabulated integral $\int_0^\infty x^ne^{-\alpha x} \, dx=\frac{n!}{\alpha^{n+1}}$ [closed]
I ran into this problem where I needed to use the following integral equality in my physics textbook.
$$\int_0^\infty x^ne^{-\alpha x} \, dx=\frac{n!}{\alpha^{n+1}}$$
where $n$ is a positive integer ...
1
vote
3answers
105 views
Prove that $(a b+b c+c a-1)^{2} \leq\left(a^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right)$.
Let $a$, $b$, and $c$ be real numbers. Prove that $$(a b+b c+c a-1)^{2} \leq\left(a^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right)\,.$$
In solution of this author take Let $a=\tan x, b=\tan y, ...
3
votes
0answers
87 views
Need some help with solving this integral…
I have to solve the following integral as part of another bigger expression:
$$
\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]\...
1
vote
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 ...
1
vote
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 ...
0
votes
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:
$ \...
2
votes
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
votes
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
$$\dfrac{u^3}{(u^2+1)^3}=\dfrac{Au+B}{(u^2+1)}+\dfrac{Cu+D}{(u^2+1)^2}+\dfrac{Au+B}{(u^2+1)^3}$$...
0
votes
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
votes
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
votes
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 $e^{-x}=t$, $-e^{-x}\ dx=dt$, $dx=-\dfrac{dt}{t}$
$$\int\dfrac{1}{\sqrt{5t^2+4t+1 } }\left(-\...
0
votes
5answers
125 views
How do I evaluate $\int{\frac{x^3}{x^2-1}dx}$ using trigonometric substitution?
How do I use trigonometric substitution to rewrite and evaluate
$$\int{\dfrac{x^3}{x^2-1}dx}?$$
I have no trouble using long division of $x^3$ by $x^2-1$ which is $x+\dfrac{x}{x^2-1}$ to get the ...
2
votes
1answer
26 views
Maximising the argument of $\sin and \cos$, given a linear relation between them
Here is a problem I have been struggling with for a while,
If $$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
votes
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 ...
