Questions tagged [substitution]
Questions that involve a replacement of variable(s) in an expression or a formula.
0
votes
2answers
18 views
Are these legitimate rules/formula for integration without using the substitution method?
I'm talking about:
$\int(ax+b)^ndx=\frac{(ax+b)^{n+1}}{(n+1)(a)}$
$\int\frac{1}{ax+b}dx=\frac{1}{a}ln(ax+b)$
$\int e^{ax+b}dx=\frac{e^{ax+b}}{a}$
$\int a^{ax+b}dx=\frac{a^{ax+b}}{(ln|a|)(a)}$
Of ...
1
vote
0answers
11 views
Gaussian integral and polar change of variable
I would like to compute the Gaussian integral $\int_{-\infty}^{+\infty}e^{-x^2/2} dx$ using a polar substitution.
I know the usual proof using Fubini's theorem and the polar substitution of variables....
0
votes
1answer
17 views
How to transform this Diffusion equation ?
I have a diffusion equation:
$$ \nabla^2C = \frac{\partial C}{\partial t}$$.
Now I would like to transform this equation into a co-ordiante frame which movies with the unperturbed palaner ...
0
votes
0answers
21 views
Integration of $g(x) = \begin{cases} x^{1/c-1}e^{-x^{-c}} &x>0\ \ \\ 0 &\text{else} \end{cases} \text{and} \ c\neq 0$
I need help to integrate this function:
$$\begin{align}
&g(x) = \begin{cases}
x^{1/c-1}e^{-x^{-c}} \ &x>0\ \ \\ 0 &\text{else} \end{cases}\\[5pt]
&\text{and} \ c\neq 0
\end{align}...
0
votes
1answer
17 views
Solve for x negative exponential
I have the following equation:
$y = 14857x^{-1.092}$
I know $y = 43$, how do I rewrite this to solve for $x$.
e.g. $43 = 14857x^{-1.092}$
0
votes
4answers
56 views
Methods to solve $\int_{0}^{\infty} x^{n}\cos(x)\:dx$
I've been playing around with the following integral and was wondering if it can be generalised to any Real $n$. Does anyone know of any methods to approach this one?
$$ I = \int_{0}^{\infty} x^n \...
1
vote
2answers
57 views
Solving $\frac{dy}{d\theta} + y \cos \theta = \frac{1}{2} \sin 2\theta$
Solve
$$\frac{dy}{d\theta} + y \cos \theta = \frac{1}{2} \sin 2\theta$$ with $y(\pi/2) = 4$.
I couldn't think of any useful substitution, so I tried simplifying this DE, but didn't get anywhere. ...
3
votes
2answers
33 views
How can a doubly improper integral become a singly improper integral after substitution?
How can a doubly improper integral become a singly improper integral after substitution?
If we take $x=\sin^2{t}$ then:
$$\int_0^1\frac{1}{x\sqrt{1-x}}dx=2\int_0^{\pi/2}\csc{t}dt$$
On the lefthand ...
2
votes
4answers
35 views
solving an inseparable differential equation involving exponentials
Solve $2xe^y$ + $e^x$ + ($x^2$ + 1)$e^y$$\frac{dy}{dx}$ = 0 with $y$ = 0 when $x$ = 0.
So this is clearly an inseparable differential equation so I thought the standard way to approach this was with ...
1
vote
3answers
45 views
Solving an inseparable initial value problem
Solve $t$$\frac{dx}{dt}$ = $x$ + $\sqrt{t^2 +x^2}$ with $x$(1) = 0.
I tried to use substitution of $t = ux$ and ended up getting down to ( $\frac{1}{u\sqrt{u^2+1}}$ + $\frac{1}{u}$ ) $du$ = -$\frac{1}...
2
votes
1answer
100 views
How to derive the substituted Partial Differential Equation
I have a second order partial differential equation.
$\frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial z^2} + \frac{2}{l}\frac{\partial U}{\partial z}=0$.
I need to introduce a ...
1
vote
1answer
50 views
How isolate y from $a=\sqrt{(y^2+x^2)^3} + \sqrt{(y^2-x^2)^3}$
$$a=\sqrt{(y^2+x^2)^3} + \sqrt{(y^2-x^2)^3}$$
I need to isolate y in a family of such expressions.
Can someone give me a guide to learn how deal with those...
There's no further information or ...
0
votes
1answer
24 views
How to express this simple equation in terms of a different variable?
I am using the equation:
$$y=(1-e^{-ax})*d$$
For the shape of a logarithmic attack phase of an audio synthesizer envelope. The idea is for it to go from (0,0) to (t,1) at a given time (x=t) value.
...
2
votes
1answer
68 views
$\int\sqrt{1-\tan x}~\mathrm{d}{x}.$ (Integral of a trigonometric function under square root)
$\int\sqrt{1-\tan x}~\mathrm{d}{x}.$ is an integral which I am not able to solve. I have restricted my ideas on trigonometric substitution but cannot conclude to an answer...will really appreciate if ...
1
vote
2answers
49 views
How Isolate y from $a=\sqrt{(y^2+(a+x)^2)^3}$ [closed]
Just that!
I'm asking for a method to isolate $y$ from such expressions:
$$a=\sqrt{(y^2+(a+x)^2)^3}$$
or even easier:
$$a=\sqrt{(y^2+x^2)^3}.$$
EDIT:
I'm just trying to revisit some of the basics ...
0
votes
1answer
19 views
Renaming integration variable in Fourier transformation
In my lecture on quantum field theory, we have recently discussed the canonical quantisation of a scalar field. There is one particular calculation, that I do not quite understand.
Namely, we have ...
0
votes
1answer
20 views
Verification by direct substitution using matrices of ODE
I have this matrix:
$$
\overline X=\begin{pmatrix}
c_1e^{-2t} +2c_2e^{5t} \\
-3c_1e^{-2t}+c_2e^{5t} \\
\end{pmatrix}
$$
I want to verify that the above is a soltuion to
$$
\...
0
votes
1answer
101 views
Olympiad Inequality with Condition
I would like to prove this :
Let $x,y,z$ be positive real numbers such that $xyz=1$ then we have :
$$\frac{\left(\frac{1}{x}+\frac{z}{x}+z\right)\left(1+\frac{1}{x}+z\right)}{3\left(\frac{z}{x}\...
0
votes
3answers
66 views
Prove $\int f(x)f'(x)\,dx = \frac {[f(x)]^2}2 + c$ through substitution
I've always taken integration by substitution for granted but recently I've learned that differentials can't fully be treated as variables and that the process of integration by substitution is really ...
0
votes
5answers
53 views
Integration question (don't know what to substitute).
How do you integrate the following?
$$\int\left(\frac{1+x}{1-x}\right)^{\frac{1}{2}}dx$$
Please do not give the full answer, but just what to start with. Thanks.
0
votes
2answers
56 views
Solve $\int_0^{\infty} x^2 \frac{x}{\theta}e^{\frac{x^2}{2\theta}} dx $
Solve $$\int_0^{\infty} x^2 \frac{x}{\theta}e^{\frac{x^2}{2\theta}} dx $$
Solution:
$$\int_0^{\infty} x^2 \frac{x}{\theta}e^{\frac{x^2}{2\theta}} dx = - \int_0^\infty x^2 de^{\frac{-x^2}{2\theta}}dx^...
1
vote
0answers
52 views
Find the integral of the following function.
Question.
My approach:
Unfortunately I am stuck right after this step.
I cannot determine what kind of "substitution" would bring the denominator as in the whole function in any solvable form like;
...
0
votes
1answer
28 views
How to use algebra to solve this?
Use Algebra to solve for $x$: $8(9^x)+3(6^x)-81(4^x)=0$
My attempt was to decompose the exponential functions into $2^x$ and $3^x$ where possible and then substitute them for some $p$ and $q$. I was ...
2
votes
4answers
84 views
Let $a, b, c \in \mathbb{R^+}$ and $abc=8$ Prove that $\frac {ab+4}{a+2} + \frac {bc+4}{b+2} + \frac {ca+4}{c+2} \ge 6$
Let $a, b, c \in \mathbb{R^+}$ and $abc=8$ Prove that $$\frac {ab+4}{a+2} + \frac {bc+4}{b+2} + \frac {ca+4}{c+2} \ge 6$$
I have attempted multiple times in this question and the only method that I ...
1
vote
2answers
63 views
Find maximum value
Given $0 \leq a,b,c \leq \dfrac{3}{2}$ satisfying $a+b+c=3$. Find the maximum value of $$N=a^3+b^3+c^3+4abc.$$
I think the equality does not occur when $a=b=c=1$ as usual. I get stuck in finding the ...
3
votes
2answers
86 views
Maximize the value of $\sqrt{x-x^2}+\sqrt{cx-x^2}$ without using calculus
Assume that $c$ is positive. How can we maximize the value of $\sqrt{x-x^2}+\sqrt{cx-x^2}$ with respect to $x$ without the use of calculus?
With calculus, we can easily find out that the max of the ...
4
votes
2answers
63 views
Calculus u-substitution: Choosing between $\frac{dx}{du}$ and $\frac{du}{dx}$
I searched and couldn't find any answers to this one. It's probably a dumb question but one that has been troubling me.
Let's say I have an integration:
$$\int \frac{1}{3x+2}$$
It seems the correct ...
4
votes
1answer
55 views
Logarithmic equation $\log_2(x+4)=\log_{4x+16}8$
So the problem goes:
What is the product of all solutions in the equation
$$\log_2(x+4)=\log_{4x+16}8$$
The solution to this should be $31\over4$, but I got $-14$. This is what I did:
\begin{...
2
votes
3answers
63 views
For u-substitution of a definite integral, is it acceptable to have the limits (bounds) of the new integral equal to one another?
For example, I have the problem:
$$
\int_{-1}^1x^4\left(1-x^2\right)^2dx
$$
I set $\,u=1-x^2\,$, which gives $\,x^2=1-u\,$, $\,x=\pm\sqrt{1-u}\,$ and $\,-\frac{1}{2}du=xdx$.
For the upper and lower ...
2
votes
1answer
39 views
How do you substitute an arbitrary conic equation with trigonometric functions?
For example,
Circle ($\frac{x^2}{a^2}+\frac{y^2}{a^2}=1$) : $$ \left\{
\begin{array}{c}
x\to a\cos \theta \\
y\to a \sin \theta \\
\end{array} \right.
$$
Oval ($\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$) : ...
1
vote
4answers
60 views
Integration by substitution vs. polynomial expansion first…different results??
I came across a textbook problem that showed an integral solved with the substitution method:
$$\int_a^b(b-x)^2dx = \left(-\frac{(b-x)^3}{3}\right)\Biggl\vert_a^b$$
I then attempted to solve the ...
1
vote
1answer
34 views
Integration by substitution, finding a new formula in terms of the substiution
I am trying to solve a question that involves integration by substitution.
Let $\displaystyle I = \int^{1}_{0}\frac{\sqrt{x}}{2- \sqrt{x}}dx$
Using the substitution $u = 2-\sqrt{x}$ show that $...
0
votes
1answer
60 views
How to prove the given inequality using CS or AM GM HM inequality [closed]
I was solving a problem on triangular inequalities and I have ended up with the following inequality required to be proven
$$\sum_{cyc}\frac{x}{2x+y+z}\geq\frac{9\sqrt{3(x+y+z)xyz}}{4(x+y+z)^2},$$
...
0
votes
0answers
39 views
Matrix Operations and Dependent Variables
Consider the following systems along with the corresponding substitutions:
\begin{align} \text{System 1} \\
2x^2 + 3y + 5z^z & = 7 \\
4x^2 + 9y + 10z^z & = 2 \\
x^2 + y = 2z^z & = 1
\end{...
4
votes
3answers
137 views
Roots of $2x^3-4x+1$
I'm having difficulty getting the solution to the cubic equation $2x^3-4x+1=0$ and from http://www2.trinity.unimelb.edu.au/~rbroekst/MathX/Cubic%20Formula.pdf it claims that the general solution to $...
1
vote
1answer
24 views
Why can you substitute both the real and imaginary parts of an equation separately into another?
The title is very vague, apologies. I'm taking A Level Further Mathematics, and we've come to the end of our topic on Argand Diagrams.
There is a question (which I'm not looking for the answer to), ...
4
votes
1answer
111 views
Prove: for all $x \in (0, 1], 2^x+2^{\frac{1}{x}} \leqslant 2^{x+\frac{1}{x}}$
This problem is from my math teacher.I tried using Calculus, the derivative function is like a black hole.Then I graphed it by Mathematica. As the following picture shows, I was strongly astonished. ...
-1
votes
3answers
43 views
Non-linear first order ODE with auxiliary variable
I have a problem with this equation: $y'(x)=\frac{2y(x)-x}{2x-y(x)}$.
Using $y=xz$ i'm arrived to prove that $\frac{z-1}{(z+1)^{3}}=e^{2c}x^{2}$, but now i'm stuck.
How can i explain the $z$? I've ...
0
votes
0answers
34 views
Replacing known “things” in differential equations
So I have the following differential equation:
\begin{equation}
\ddot x + \frac{b}{m}\dot x + \frac{2k}{m}x = 0
\end{equation}
Let's say I also find that $a(t) = -\ddot x$, which helps me to reach ...
2
votes
1answer
39 views
Legitimate change of variable in integral of an integral function?
Consider the following expression:
$$ A^q=\int_{\mathbb{R}^{n}}\left(\int_{\mathbb{R}^{n}}\left|f\left(x+y\right)g\left(x+y+Cy\right)\right|^{p}dy\right)^{q/p}dx, $$ where $1\leq p,q<\infty$, $C$ ...
6
votes
0answers
127 views
What is more historically correct?
I was wondering what was formally correct, when resolving an integral by substitution, use as a new variable $t$ or $u$? I noticed that here in Italy we use $t$, while I saw that abroad (e.g USA) use $...
3
votes
4answers
57 views
Non-linear second order ODE
I have to solve $$ y''(x)+(y'(x))^2=y'(x). $$
Using $ y'(x)=z $, I can write $$\int \frac{1}{z-z^2}dz=\int dx $$
So:
$$\frac{1}{z(1-z)}=\frac{A}{z}+\frac{B}{1-z}$$
leads to
$$ \int \frac{1}{z(1-z)...
2
votes
1answer
87 views
How to prove inequality $(a^2+b^2+c^2)^3\ge6(a^3+b^3+c^3)^2$ when $a+b+c=0$?
I know the identity $a^3+b^3+c^3-3abc = 1/2 (a+b+c) [(a-b)^2+(b-c)^2+(c-a)^2]$.
So when it comes to this problem where $a+b+c=0$, you get $(a^2+b^2+c^2)^3\ge54(abc)^2$ when $a+b+c=0$
(because $ a^3+b^...
0
votes
1answer
32 views
unknown square root substitution in integration
The integral
$$
\int d\theta=\int\frac{r_0}{R}\sqrt{\frac{R^2-r^2}{r^2-r_0^2}}\frac{dr}{r} \tag{1}
$$
appears in the solution of Venezian to the terrestrial brachistochrone problem. (Venezian, ...
1
vote
3answers
51 views
Substitution in $\int \cos^3(x)$
I have an problem with the substitution in
$$\int \cos^3(x)dx$$
Ive seen some answers where the integration is done by substituting
$$\int \cos^3(x) = \int \cos(x)(1-\sin^{2}(x))$$
$u = \sin(x)$
...
2
votes
1answer
61 views
Given triangle sides $a$, $b$, $c$, show $\sum_{\text{cyc}}\frac{a}{a+b}\geq\frac32\prod_{\text{cyc}}(\frac{a}{a+b}+\frac{b}{b+c})$
If $a,b,c$ are three sides of a triangle then I need to prove that
$$
{\frac {a}{a+b}}+{\frac {b}{b+c}}+{\frac {c}{c+a}}\geq \frac{3}{2}\, \left( {
\frac {a}{a+b}}+{\frac {b}{b+c}} \right) \left( {\...
2
votes
4answers
95 views
Prove that if in a tetrahedron if two pairs of opposite edges are perpendicular then the third pair is also perpendicular.
Prove that if in a tetrahedron if two pairs of opposite edges are
perpendicular then the third pair is also perpendicular.
Method:
$\vec a+ \vec b + \vec c + \vec d +\vec e + \vec f = 0 \tag0$ ...
2
votes
1answer
74 views
Show that $\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$ for a triangle with sides $a$, $b$, $c$ and area $S$
Let be $a$, $b$, $c$ sides of a triangle and $S$ his area. Prove that $$\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$$
My idea: $b^2+c^2-a^2 = 2bc \cos A$, so the inequality is ...
0
votes
0answers
14 views
check of calculation of a surface integral
Find surface integral $$I= \int _M (x+y+z) \mathrm{d} S$$, where M is the upper half-sphere given explicitly as $z=\sqrt{a^2-x^2-y^2}$, $x^2+y^2 < a^2$.
I would like you to check if my ...
0
votes
1answer
20 views
System of 2nd order diff equations with $x_1(t), x_2(t)$
Could you help me with the following system of two equations?
I have browsed through the forum, but I haven't found a similar problem.
$$m_1x_1''(t) + k(x_1 - x_2)+ cx_1'(t) = 0$$
$$m_1x_2''(t) + k(...
