Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [substitution]

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

In mathematics, the operation of substitution consists of replacing all the occurrences of a free variable appearing in an expression or a formula with 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.

2066 questions
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How is the Taylor expansion for $f(x + h)$ derived?

According to this Wikipedia article, the expansion for $f(x\pm h)$ is: $$f(x \pm h) = f(x) \pm hf'(x) + \frac{h^2}{2}f''(x) \pm \frac{h^3}{6}f^{(3)}(x) + O(h^4)$$ I'm not understanding how you are left with $f(x)$ terms on the right hand side. I…
badjr
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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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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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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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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…
math110
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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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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 +…
Skunk Works
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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,…
user420309
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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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How to evaluate $\int{\frac{\sqrt{x}}{1 + \sqrt[3]{x}}}dx$?

$$\int{\frac{\sqrt{x}}{1 + \sqrt[3]{x}}}dx$$ When I first came across the above integral I noticed it is unlikely an immediate $u$-sub as it involves both square and cube roots. So, the regular substitutions for evaluating $\sqrt{f(x)}$ like…
Dstarred
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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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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…
user167045
15
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2 answers

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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How to solve $\frac{dy}{dx}-1=\frac{x^2}{y}$

How to solve $$\frac{dy}{dx}-1=\frac{x^2}{y}$$ I tried letting $y^2=u$ which gives $$y\frac{dy}{dx}=\frac{1}{2}\frac{du}{dx}$$ So the equation is now $$\frac{du}{dx}-2\sqrt{u}=2x^2$$ Any help?
Ekaveera Gouribhatla
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Reconciling measure theory change of variables with u-substitution

In measure theory we learn that $$ \int_\Omega g \circ f d\mu = \int_{f(\Omega)}g d(\mu \circ f^{-1}) $$ where $(\Omega, \mathscr F, \mu)$ is a measure space and $f$ and $g$ are measurable. Now in calculus we have that $$ \int_{\phi(a)}^{\phi(b)}…
alfalfa
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