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Questions tagged [substitution]

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

4
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6answers
77 views

What's the answer to $\int \frac{\cos^2x \sin x}{\sin x - \cos x} dx$?

I tried solving the integral $$\int \frac{\cos^2x \sin x}{\sin x - \cos x}\, dx$$ the following ways: Expressing each function in the form of $\tan \left(\frac{x}{2}\right)$, $\cos \left(\frac{x}{2}\...
2
votes
1answer
90 views

Stuck with Integration by Substitution

I have a question where we need to find an integral using where "$u = 1+e^{x}$" for the equation "$\int \frac{e^{3x}}{1+e^{x}}dx$". However when I substitute it I end up with "$\int \frac{(u-1)^{3}}{...
-1
votes
2answers
63 views

Very Hard System of Equations

Solve the system of equations: \begin{cases} \sqrt{xy}(x + 3y)(3x + y) = 14 \\ (x + y)(x^2 + y^2 + 14xy) = 36. \end{cases} Suppose $x + y = m$ and $xy = n$. So I get \begin{cases} (3m^2+4n) \sqrt ...
3
votes
3answers
53 views

Prove $\frac{1}{3}(a+b+c)^2\leq a^2 + b^2 + c^2 + 2(a-b+1).$

Prove that for $a>1$,$b>1$ and $c>1$ where $a,b,c\in \mathbb{R}$ $$\frac{1}{3}(a+b+c)^2\leq a^2 + b^2 + c^2 + 2(a-b+1).$$ My attempt: it is not so clear why is $a>1$, $b>1$ and $c>...
2
votes
2answers
59 views

Primitive of a function with $\sin \frac{1}{x}$

I have the next integral: $$\int\biggl({\frac{\sin \frac{1}{x}}{x^2\sqrt[]{(4+3 \sin\frac{2}{x})}}}\biggr)\,dx ,\;x\in \Bigl(0,\infty\Bigr)$$ I used the substitution $u=\frac{1}{x}$ and I got $$-\int\...
-2
votes
0answers
35 views

Finding the integral using u substitution, then solving for C [closed]

Find $f(x)$ given $f(\frac19)=0$ and $$f'(x)=\frac{1}{\sqrt{3x}(3\sqrt x -1)^\frac 23}$$ I have made multiple attempts at this problem and believe I messed up with integrating du back into the ...
0
votes
1answer
30 views

Inequality triangle Radon substitutions

I have this inequality: $$\sum \frac {a^3}{p-a}\geq 8(2R-r)^2$$ I have tried using Radon substitutions and I get this: $$\sum \frac{(y+x)^3}{x}\geq 8(2R-r)^2$$ I know from Holder that : $$\sum \frac{(...
3
votes
1answer
94 views

Solving $8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2}$

Solve the equation $$8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2}$$ I have this idea: set $$\sqrt{x+2}=a , x+2=a^2 , \sqrt{x-1}=b.$$ So $$x-1=b^2 , 2a^2+6b^2 =8b-4$$ and $$x^2+x-2 =a^2b^2$$ and ...
1
vote
1answer
68 views

Inequality. $\sum_{cyc}(\frac{1}{a+b+\sqrt{2a+2c}})^3 \le \frac{8}{9}$

Problem. When $a, b, c>0, a, b, c \in \Bbb R, 16(a+b+c)\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$, Prove that $$\sum_{cyc}(\frac{1}{a+b+\sqrt{2a+2c}})^3 \le \frac{8}{9}$$ My approach: If we let $x=a+...
0
votes
1answer
55 views

Prove the inequality $\sum_{cyc} {{a+abc} \over {1+ab+abcd}} \ge {{10} \over {3}}$ with Cauchy-Schwarz [closed]

Problem: If $abcde = 1$, $a, b, c, d, e > 0$, $a, b, c, d, e \in \Bbb R$, prove that $\sum_{cyc} {{a+abc} \over {1+ab+abcd}} \ge {{10} \over {3}}$ First I proceeded with Cauchy-Schwartz ...
0
votes
4answers
69 views

Integral utter confusion with substition and dx/du

I need to find the indefinite integral I = $$\int e^x (1+e^x)^{\frac{1}{2}}$$ by using a proper substition method. I tried it on https://www.integral-calculator.com and it gave the following ...
1
vote
0answers
37 views

Transformation Jacobian for integration by substitution in single and multiple variables?

I wonder how to reconcile the transformation rules for integration by substitution in a single variable vs several variables. In the single-variable case the Jacobian factor $\varphi'(x)$ is just a ...
1
vote
0answers
60 views

Introducing a new variable for evaluating integrals

In some problems we use reverse substitution for evaluating integrals . For example consider $\int \sqrt{1-x^2}dx$ and $x = \sin(t) \ , \ \frac{-\pi}{2}\le t \le \frac{\pi}{2}$ . In this case , ...
4
votes
2answers
73 views

What substitution do I use to integrate this?

I don't know how to proceed in this integration. $$\int \frac{d \theta} {\sqrt{3 + 2 \cos \theta}} $$ I could think of two substitutions: $3 + 2 \cos \theta = t^2$ $\cos \theta = \frac{\sqrt{3}} {\...
6
votes
1answer
42 views

Integrating rational function by multiplying with a high-power polynomial term

When evaluating the integral $\int \frac{x^2 + 8}{x^3 + 9x} \textrm{d}x$, WolframAlpha gave an interesting step-by-step suggestion. Here is the suggested method: $$\int \frac{x^2 + 8}{x^3 + 9x} \...
3
votes
1answer
70 views

Sine from negative cosine

I'm trying to solve the integral $I=\int \frac{dx}{x^3 \sqrt{x^2-4}}$ ($x<0$) which can be solved by substituting $x=2 \sec t$. One ends up with $$I=-\frac{1}{16} \left( t+\sin t \cos t \right)+C$$ ...
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votes
0answers
20 views

Help with parameters when computing volume

I have just started computing volume of objects confined by planes using Fubini and substitution and I have come across a problem that I can't figure out what new parameter to choose. the planes are: ...
1
vote
2answers
57 views

Solve the integral $\int_0^1\int^1_xy^4e^{xy^2}dydx$.

Solve the integral $\int_0^1\int^1_xy^4e^{xy^2}dydx$. I think that variables substituation is neede here. I've substitute $$ \\ \left\{\begin{matrix} u=xy^2\\ v=y \end{matrix}\right. \ $$ and ...
1
vote
2answers
80 views

How to integrate $\frac{\cos(x)}{x}$ using substitution

Trying to integrate $$\int \frac{\cos(x)}{x} dx = \int \frac{1}{x}\sin'(x) dx$$ by substituting $\sin(x)$, but it either becomes more complicated or I end up with a $\frac{1}{x}$ still in the integral....
1
vote
2answers
37 views

Why can we redefine the definition of a variable during substitution? Or let say assume it has two different values

This concept I have asked a few people, but none of them are able to help me understand, so hope that there's a hero can save me from this problem!!! My question occurs during substitution process, ...
0
votes
3answers
79 views

Applying substitution to $\int \sqrt{1+\sin(x)}$ [duplicate]

I'm having a problem with substituting when it comes to trig functions. The integral is:$$\int\sqrt{1+\sin(x)}dx$$ Substituting: $1+\sin(x) = u \implies \frac{d}{dx} u = \cos(x) \implies dx = \frac{...
2
votes
3answers
39 views

Find the exhaustive values of “a”, given the condition

If the equation $2^{2x} + a*2^{x+1} + a + 1=0$ has roots of opposite sign then the exhaustive values of a are? I tried taking $2^x = t$. But then didn't know what to do. The equation became, $t^2 + ...
0
votes
1answer
68 views

Substitution - mistake

Where do I mistake please? My computation differ from the result in the text about red terms. Thank you
0
votes
1answer
45 views

How to define substitution using ZFC

One question I've had regarding ZFC is how to define substitution. I cannot see how it's possible, despite the frequent use of substitution within both pure and applied mathematics. Just to be clear, ...
2
votes
2answers
56 views

Does the constant $C$ in this solution to a differential equation equal infinity?

The problem is $y' = -\frac{1}{t^2} - \frac{1}{t}y + y^2;\ y_p = \frac{1}{t}$. My solution is $$\begin{align} y = \frac{1}{t} + B &\implies y' = -\frac{1}{t^2} + B' \\ &\implies -\frac{1}{t^...
0
votes
2answers
51 views

Substitution in an Integral from log to inverse

Considering equations (2) (8) and (9) below. Please, I am wondering what is the insight into how the substitution was made to get (9). I have exhausted much energy trying to figure this out. Note ...
1
vote
2answers
68 views

How to solve $T(n) = T(2n/3) + \lg^2 n$ by substitution?

My solution through substitution is as follows: $$T(n) = T(2n/3) + \lg^2 (n)$$ $$T(2n/3) = T(4n/9) + \lg^2 (2n/3)$$ $$T(4n/9) = T(8n/27) + \lg^2 (4n/9)$$ And so on... But my actual problem is how ...
2
votes
1answer
168 views

Solving $ \left(\frac{ ( x^{3}+1 )^{3}+8 }{16}\right) ^{3}+1=2x$

Solving : $$\left(\frac{ ( x^{3}+1 )^{3}+8 }{16}\right) ^{3}+1=2x$$ My Try : $$\left(\frac{ ( x^{3}+1 )^{3}+8 }{16}\right) ^{3}=\left(\dfrac{(x+1)^3+2^3}{2^4}\right)^3$$ $$ x^3+a^3=(x+a)^3-3ax(x+...
0
votes
2answers
60 views

Help needed in understanding a example of complex numbers.

I was reading an example from the chapter complex numbers. Here it is: Prove that $${\sqrt{7\over2} }≤ |1 + z|+|1 − z + z^2| ≤ 3{\sqrt{7\over6}}$$ for all complex numbers with $|z| = 1$. ...
1
vote
1answer
47 views

Inequality Question - Homogeneity

Let $n>3$ and $x_1, x_2, \ldots, x_n$ be positive real numbers with $x_1x_2\cdots x_n=1$. Prove that $$ \frac{1}{1+x_1+x_1x_2}+\frac{1}{1+x_2+x_2x_3}+\cdots+\frac{1}{1+x_n+x_nx_1}>1. $$ I’d ...
3
votes
1answer
119 views

If $ab+bc+ca=3$ for non-negative $a$, $b$, $c$, show that $\sum_{cyc}a^2b^2+\sum_{cyc}\frac{12a^2b^2c^2}{(a+b)^2}\ge 12abc$

Problem Let $a,b,c\ge 0$,and such $ab+bc+ca=3$, show that $$\sum_{cyc}a^2b^2+\sum_{cyc}\dfrac{12a^2b^2c^2}{(a+b)^2}\ge 12abc\tag{1}$$ A few hours ago, I asked for an error inequality. Wrong ...
9
votes
0answers
78 views

Is there a substitution which transforms every Fermat curve into an elliptic curve?

A Fermat Curve of degree $n$ is the set of solutions to $x^n+y^n=z^n$, $x,y,z\in \mathbb R$. In this question, the OP provides a substitution which relates a Fermat Curve of degree $n=3,4$ to two ...
1
vote
4answers
58 views

Integral by trigonometric substitution: $ \int \:\frac{2y^3}{\sqrt{1-y^2}}\mathrm dy $

$$ \int \:\frac{2y^3}{\sqrt{1-y^2}}\mathrm dy $$ I substitute $$ y = \sin t $$ then $$ 2 \int \:\frac{\sin^3 t \cdot \cos t}{\cos t}\;\mathrm dt = 2 \int \sin^2 t\;\mathrm d \cos t $$ What should I ...
3
votes
4answers
61 views

Use the spherical coordinates to compute the integral $\int\limits_{B} z^2 dx dy dz$ where B is defined by $1\leq x^2 + y^2 + z^2 \leq 4$

, however the answer I got to is different than the answer sheet. The answer sheet says that it should be $\frac{62}{15}$ Am I making some mistake or is the answer sheet incorrect?
3
votes
1answer
92 views

Need help computing the double integral $\int_{0}^{\infty} \int_{0}^{\infty} \frac{f(x + y)}{x + y} \mathop{dy} \mathop{dx}$

Need help computing the double integral $\int_{0}^{\infty} \int_{0}^{\infty} \frac{f(x + y)}{x + y} \mathop{dy} \mathop{dx}.$ I know that $\int_{0}^{\infty} f(u) \mathop{du}$ equals $1$. The entire ...
1
vote
1answer
23 views

Examples of higher order nonlinear ODEs that can be solved via substitution

Background I'm currently reviewing for the final exam of my undergraduate Ordinary Differential Equations course, and a question on the last exam that stumped me was the following: Find the ...
1
vote
1answer
77 views

Calculate integral with two exponential functions

If $c >0$ and $d >0$, show that $$\int\limits_{0}^\infty e^{-cx} \frac{d}{x^{\frac{3}{2}} \sqrt{2 \pi}} e^{-\frac{d^2}{2x}} \, \mathrm dx = e^{-d\sqrt{2c}}.$$ Obviously we have that $$\int\...
0
votes
0answers
24 views

Proof of multivariable change of variables integration

Seeing the simplicity of the proof of the u-substitution integration formula ( see here ) , I would like to know if there is such an easy proof for the multivariable case with the jacobian determinant....
2
votes
1answer
45 views

Why is it that, when using the u-substitution rule, you apply the the function u to the upper and lower limits?

I was taught that we apply the function u to the upper and lower bounds of the definite integral, then use those as the new upper and lower bounds to evaluate the integral. Indeed, this works and is ...
0
votes
0answers
26 views

FOL: substitution of open formulas and the Substitution Theorem in Van Dalen's “Logic and Structure”

Theorem 3.5.8 (Substitution Theorem), p. 72 in Van Dalen's Logic and Structure (5th ed.) states that $$\models(\varphi \leftrightarrow \psi) \rightarrow(\sigma[\varphi/P] \leftrightarrow \sigma[\psi/...
0
votes
1answer
23 views

What is the exact criteria for when you can't evaluate an integral without using the substitution rule (only using the two parts of the FTC)?

For example, you can't integrate $2x\sqrt{1+ x^2}$ just using the two parts of the fundamental theorem of calculus. I can see that I can't solve it, but I'm not sure which unique characteristic of ...
1
vote
1answer
40 views

Change of variables in differential equation?

I have the following formula: $$f(x) = \frac{d^2w(x)}{dx^2}$$ Now I would like to normalize $x$ by dividing it by $L$? This would be the substitution: $$\hat{x}=x/L$$ How would my formula change? (...
0
votes
0answers
29 views

substitution vocabulary

I am crafting materials for college students in my classes to address algebra misconceptions, and I need a simple set of vocabulary to distinguish between two things when performing complex ...
0
votes
1answer
72 views

How can we visualise integration by substitution?

When we integrate a function with respect to dx, we are breaking the area under the function into many pieces of width dx and finding the area of each piece and then take the limit of the sum as dx ...
0
votes
0answers
20 views

Substitution to get rid of cubic terms in a two-variable expression

Consider the quadratic form $$Q(u,v) = au^4 + bu^3v+cuv^3+du^2v^2+ev^4$$ If $a,b,c,d = 1$ then we can get rid of the cubic terms by substituting $u:=x-y, v:=x+y$. But what would be the general ...
2
votes
2answers
65 views

Is integration by substitution always a reverse of the chain rule?

To integrate $\int x^3\sin(x^2+1)dx$, I took the following approach: \begin{align*} \begin{split} \int x^3\sin(x^2+1)dx&=\int x^3\sin(u)\cdot\frac{1}{2x}du\\ &=\frac{1}{2}\int x^2\sin(u)du\\ &...
1
vote
1answer
37 views

how to choose the sabstitution for Euler's integrals?

I currently have, and have to calculate the Gamma function: $$\int_2^4 \sqrt[4]{(x-2)(4-x)^3}\,\mathbb{d}x$$ As per definition gamma function is: $$\int_0^1t^{z-1}e^{-t}\,\mathbb{d}t$$ Do I ...
2
votes
3answers
42 views

Riccati differerntial equation

My book suggest to use the substitution $y(t)=y_{1}(t)+u(t)$ for Riccati odes Given the ode: $$y'=1+x^2-y^2$$ $\frac{dy}{dx}=1+x^2-y^2$ A solution is $y_{1}=x$ so $y=x+u$ $\frac{dy}{dx}=1+\frac{...
0
votes
0answers
18 views

Substitution integral problem

If in the integral, $\int (f'(x))^{-2}dx $ we make the substitution $f(x)=t$, what is the new integral? Solution of this problem (test) says is $\int t^2dt$, although I think then the integral should ...
0
votes
1answer
43 views

Substitutability in First-Order Logic

Here is a definition for substitutability found in a PDF of logic notes by Eric Pacuit: I am more concerned with the part of the definition squared in red. My question is: Given $(\forall y) \psi$, ...