Questions tagged [substitution]

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

In mathematics, the operation of substitution consists in replacing all the occurrences of a free variable appearing in an expression or a formula by a number or another expression. In other words, an expression involving free variables may be considered as defining a function, and substituting values to the variables in the expression is equivalent to applying the function defined by the expression to these values. A change of variables is commonly a particular type of substitution, where the substituted values are expressions that depend on other variables. This is a standard technique used to reduce a difficult problem to a simpler one. A change of coordinates is a common type of change of variables. However, if the expression in which the variables are changed involves derivatives or integrals, the change of variable does not reduce to a substitution.

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Why is this integration method not valid?

Let $$I=\int \frac{\sin x}{\cos x + \sin x}\ dx \tag{1}$$ Now let $$u=\frac{\pi}{2} - x \tag{2}$$ so $$I=\int \frac{\sin (\frac{\pi}{2} - u)}{\cos (\frac{\pi}{2} - u)+\sin (\frac{\pi}{2} - u)}\ du \tag{3}$$ $$=\int\frac{-\cos u}{\sin u + \cos u} \…
LJD200
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Solving $ax^3+bx^2+cx+d=0$ using a substitution different from Vieta's?

We all know, a general cubic equation is of the form $$ax^3+bx^2+cx+d=0$$ where $$a\neq0.$$ It can be easily solved with the following simple substitutions: $$x\longmapsto x-\frac{b}{3a}$$ We get, $$x^3+px+q=0$$ where, $p=\frac{3ac-b^2}{3a^2}$…
Learner
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Prove a trigonometric identity: $\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C=1$ when $A+B+C=\pi$

There is a trigonometric identity: $$\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C\equiv 1\text{ when }A+B+C=\pi$$ It is easy to prove it in an algebraic way, just like that: $\quad\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos…
MafPrivate
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Indefinite Integral $\int\sqrt[3]{\tan(x)}dx$

For calculating $\int\sqrt{\tan(x)}dx$, I used this easy…
user91500
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Quadratic inequality puzzle: Prove$ |cx^2 + bx + a| ≤ 2$ given $|ax^2+bx+c| ≤ 1$

I came across this problem as part of a recreational mathematics challenge on university: Suppose $a, b, c$ are real numbers where for all $ -1 \le x \le 1 $ we have $|ax^2 + bx + c| \le 1$. Prove that for all $-1 \le x \le 1 ,$ $$ |cx^2 + bx +…
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Square root of $2$ is irrational

I am studying the proof that $\sqrt 2$ is an irrational number. Now I understand most of the proof, but I lack an understanding of the main idea which is: We assume $\frac{m^2}{n^2} = 2$. Then both $m$ and $n$ can't be even. I do not understand,…
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Prove the inequality $\sqrt\frac{a}{a+8} + \sqrt\frac{b}{b+8} +\sqrt\frac{c}{c+8} \geq 1$ with the constraint $abc=1$

If $a,b,c$ are positive reals such that $abc=1$, then prove that $$\sqrt\frac{a}{a+8} + \sqrt\frac{b}{b+8} +\sqrt\frac{c}{c+8} \geq 1$$ I tried substituting $x/y,y/z,z/x$, but it didn't help(I got the reverse inequality). Need some stronger…
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How prove this inequality $\sum\limits_{cyc}\frac{x+y}{\sqrt{x^2+xy+y^2+yz}}\ge 2+\sqrt{\frac{xy+yz+xz}{x^2+y^2+z^2}}$

let $x,y,z$ are postive numbers,show that $$\dfrac{x+y}{\sqrt{x^2+xy+y^2+yz}}+\dfrac{y+z}{\sqrt{y^2+yz+z^2+zx}}+\dfrac{z+x}{\sqrt{z^2+zx+x^2+xy}}\ge 2+\sqrt{\dfrac{xy+yz+xz}{x^2+y^2+z^2}}$$ My try: Without loss of generality,we assume that…
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How to prove this inequality? $ab+ac+ad+bc+bd+cd\le a+b+c+d+2abcd$

let $a,b,c,d\ge 0$,and $a^2+b^2+c^2+d^2=3$,prove that $ab+ac+ad+bc+bd+cd\le a+b+c+d+2abcd$ I find this inequality are same as Crux 3059 Problem.
math110
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How prove $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le\frac{3}{\sqrt{7}}$

Let $a,b,c>0$,and such $$\dfrac{1}{a^2+2}+\dfrac{1}{b^2+2}+\dfrac{1}{c^2+2}=\dfrac{1}{3}$$ show that $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\le\dfrac{3}{\sqrt{7}}$$ my try:…
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Substitution Makes the Integral Bounds Equal

This seems like a really basic calculus question, which is a tad embarrassing since I'm a graduate student, but what does it mean when a substitution in a definite integral makes the bounds the same? For example, if we have some function of…
Klein Four
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Combined AM GM QM inequality

I came across this interesting inequality, and was looking for interesting proofs. $x,y,z \geq 0$ $$ 2\sqrt{\frac{x^{2}+y^{2}+z^{2}}{3}}+3\sqrt [3]{xyz}\leq 5\left(\frac{x+y+z}{3}\right) $$ Addendum. In general, when is $$…
picakhu
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Evaluating $\int_0^1 \frac{3x}{\sqrt{4-3x}} dx$

So this is the integral I must evaluate: $$\int_0^1 \frac{3x}{\sqrt{4-3x}} dx$$ I have this already evaluated but I don't understand one of the steps in its transformation. I understand how integrals are evaluated, but I don't understand some of…
Arkilo
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How prove this $x^3+y^3+z^3+3\ge 2(x^2+y^2+z^2)$

Question: let $x,y,z>0$ and such $xyz=1$, show that $$x^3+y^3+z^3+3\ge 2(x^2+y^2+z^2)$$ My idea: use AM-GM inequality $$x^3+x^3+1\ge 3x^2$$ $$y^3+y^3+1\ge 3y^2$$ $$z^3+z^3+1\ge 3z^2$$ so $$2(x^3+y^3+z^3)+3\ge 3(x^2+y^2+z^2)$$ But this is not my…
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Given three a-triangle-sidelengths $a,b,c$. Prove that $3\left((a^{2}b(a-b)+b^{2}c(b-c)+c^{2}a(c-a)\right)\geqq b(a+b-c)(a-c)(c-b)$ .

If you are interested in IMO 1983 please see: Given three a-triangle-sidelengths $a,b,c$. Prove that: $$3\left ( a^{2}b(a- b)+ b^{2}c(b- c)+ c^{2}a(c- a) \right )\geqq b(a+ b- c)(a- c)(c- b)$$ If $c\neq {\rm mid}\{a, b, c\}$, the inequality is…
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