**871 substitution questions.**

Can anyone explain me this definite integral I got from a probability problem:
$\mathbf E[X] = 2\int_0^\infty x^2e^{-x^2} dx$
$[x^2 = z]$
$=\int_0^\infty z^{{3\over2}-1}e^{-z} dz$
$= \Gamma({3\...

For $a \ge0$ find maximum of $P=$$3x\sqrt{a-y^2}-3y\sqrt{a-x^2}+4xy+4\sqrt{a^2-ax^2-ay^2+x^2y^2}$
I think maximum of P when x=-y but i donβt know how to make it reasonable

I have a challenging integral in front of me, which has evaded all types of substitutions and seems to be intractible to me(P.S.:I am a high schooler). Here it is:
$$\int_0^1 \frac{x^a-1}{ln (x)}$$
...

I know we can derive something like this from =Elim rule:
$$\varphi(a,b)$$
$$b=c$$
$$\therefore\varphi(a,c//b)$$
But I want to know if we can also go ahead and derive something like this if there was ...

This is essentially a question about an answer I received at -
Integrating a function using u-substitution. Initially, I thought I understood it, but looking at it, I do have uncertainties.
I just ...

Using substitution with $u=\sin{x}$, The integral of $\sin{x}\cos{x}$ is equal to $\frac{1}{2}\sin^2{x}$. This is the solution I require.
However, why is it that when I use the double angle formula $\...

I am given an integral:
$$\int \frac{e^{4x}}{36+e^{8x}}\,\mathrm dx $$
I am told to solve this by substitution where $u=e^{4x}/6$, and that I need to write the integrand as a function of $u$.
I am ...

Given $a$, $b$ and $c$ are positive real numbers. Prove that:$$\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$$
Additional info: We can't use induction. We should mostly use Cauchy ...

I was recently reading the book "Problems From The Book". In this book, there is a chapter on 'useful substitutions'. The techniques mentioned here often assume that $x,y,z > 0$ along with some ...

Find the largest constant $k$ such that $$\frac{kabc}{a+b+c}\leq(a+b)^2+(a+b+4c)^2$$
My attempt,
By A.M-G.M, $$(a+b)^2+(a+b+4c)^2=(a+b)^2+(a+2c+b+2c)^2$$
$$\geq (2\sqrt{ab})^2+(2\sqrt{2ac}+2\sqrt{...

Consider the following differential equation
$$
\dot{x}(t) = f(x(t))+g(u(t)), \ \ t\ge 0,
$$
where $x(0)\in\mathbb{R}$ and $u(t)$ is an external input. Suppose that the functions $f$ and $g$ are such ...

My course's notes say that for a function $u=u(x,y)$, if we define $\alpha = x+ay$, and $\beta = x+by$, then:
$u_x = u_\alpha + u_\beta$, and $u_y = au_\alpha + bu_\beta$.
And this can help solve ...

I'm trying to work out the following indefinite integral:
$\int x^3 \sqrt{1-x^2}$. To solve this, I said let $u=1-x^2$. Then, $\frac{du}{dx}=-2x$, and $du=-2x \cdot dx$, and $dx=\frac{du}{-2x}$. ...

I need to compute the integral $\int \dfrac{2x}{(x^2+x+1)^2} \cdot dx$. I tried using the integration of a rational function technique, with $\frac{Ax+B}{x^2+x+1}+\frac{Cx+D}{(x^2+x+1)^2}$, but this ...

This perhaps might be a simple question but for some reason I can't seem to figure out a satisfying answer for it.
Let's consider the following indefinite integral that does not require a ...

Prove the following inequality for non-negative real numbers $a,b,c$:
If $a+b+c=3$ then:
$$(a+b^2)(b+c^2)(c+a^2)\le13+abc(1-2abc)\qquad(1)$$
There are two more variants of the same problem. The ...

$$
\int \frac{\sqrt{x}}{\sqrt{x}-3}dx
$$
What is the most dead simple way to do this?
My professor showed us a trick for problems like this which I was able to use for the following simple example:
...

I have to find the indefinite integral: $\int \frac{1}{\sqrt{1+e^x}}\cdot dx$.
I tried substituting $u=e^x$ and then $v=1+u$, and I find that $\int \frac{1}{\sqrt{v}}=2\sqrt{v}+c=2\sqrt{u+1}+c=2\sqrt{...

I need to find the indefinite integral:
$\int \cos(\log(x)) dx$.
I've tried using u-substitution, setting $u=\log(x)$. Then I get:
$\int \cos(u)du$, where $du$ is $\frac{1}{x}dx$.
The worked ...

Problem
I have a question about variable substitution for a multivariate integral. Let me first pose the question and then provide some context, since I think the context is not necessarily very ...

I'm trying to figure out how to integrate this function. I tried several tricks from my toolkit, but I can't seem to figure it out.
$$\int\ \frac {e^{2x}-6e^x}{e^x+2}\ dx$$
So let's say that I ...

In a script I found that the indefinite integral of $\int{\frac{c}{\sin(x)\sqrt{\sin^2(x)-c^2}}dx}$ is $\sin^{-1}(\frac{\cot(x)}{c})$. I know that $\frac{1}{\sin^2(x)} = \cot^2(x)+1$. I wanted to use ...

I am trying to compute the integral $\int\frac{1}{\sqrt{x^2+cx}}dx$.
To begin I completed the square of the denominator resulting in $$\int \frac{1}{\sqrt{(x+\frac{c}{2})^2-\frac{c^2}{4}}}dx$$
I then ...

[Recall that $f^{(p)}$ denotes the $p$th derivative of $f$, and $f^{p}$ is the $p$th power of f.]
a. $\int [5π^3(π₯) + 7π^2(π₯) β 3]\cdot f'(π₯) πx$
b. $\int csc^2(3f'(x))\cdot f''(x)dx$
c. $β«\...

Prove that for a + b + c =1 and a,b,c are positive real numbers,
then
$$\frac{bc+a+1}{a^2+1} + \frac{ac+b+1}{b^2+1} + \frac{ab+c+1}{c^2+1} \le \frac{39}{10}$$
My try:
if one term is proven to be $\...

$$
\int_{0}^{2} \sqrt{x+ \sqrt{\frac{x^2}{2}+\sqrt{\frac{x^3}{3}+1}}}dx\>
$$
Wolfram alpha gives its answer as 3.01376.
Can any one provide me a solution for this?

My question is about dropping the absolute-sign when solving this integral using substitution:
$$\int\frac{dx}{x^2(x^2-1)^{3/2}}$$
We can do the following substitution: $x=\sec\theta$ and $dx=\sec\...

I'm trying to go through the solution for a SIS epidemic model, but I'm stuck figuring out the following steps.
So we have
$iβ²=r(kβi)iβΞ±i$
And to solve this, they did a substitution $y=i^{β1}$. ...

I'm trying to go through the solution for a SIS epidemic model, but I'm stuck figuring out the following steps.
So we have
$i' = r(k - i)i - \alpha i$
And to solve this, they did a substitution $y ...

If you are given
$$\int \frac{x}{\sqrt {1-x^2}}dx$$
How is it that $-\frac{1}{2}$ can be factored outside of the integral?
The exponent in the denominator is $\frac{1}{2}$, but I do not see where ...

I am trying to solve this integral:
$$\int\frac{dx}{x(3+x^2)\sqrt{1-x^2}}$$
We can use substitution:$$1-x^2=u^2$$ and $$-x dx=u du$$
Which gives us:
$$-\int\frac{udu}{(1-u^2)(4-u^2)|u|}$$
Now my ...

For a hobby project gone too far, I'm trying to find the maximum separation between two ellipses representing planetary orbits. The math has reached a point where it is officially over my head. I have ...

I have seen two approaches to the method of integration by substitution (in two different books). On searching the internet i came to know that Approach I is known as the method of integration by ...

let $a,b,c>0$ show that :$$(a^3+b^3+abc)^2(b^3+c^3+abc)^2(c^3+a^3+abc)^2-(a^2b+b^2c+c^2a)^3\cdot(ab^2+bc^2+ca^2)^3\ge 0$$Maybe there's some kind of identity in there.so I use wolfampha Calculated ...

Let $a$, $b$ and $c$ be non-negative numbers such that $ab+ac+bc\neq0$. Prove that:
$$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}+\frac{3\sqrt[3]{a^2b^2c^2}}{2(a^2+b^2+c^2)}\geq2\tag{1}$$
we know ...

Let $a,b,c>0$ and $abc=1$,show that
$$(a^{10}+b^{10}+c^{10})^2\ge 3(a^{14}+b^{14}+c^{14})$$
since
$$LHS=\sum \left(a^{20}+\dfrac{2}{a^{10}}\right)$$
it is prove
$$\sum_{cyc}\left(a^{20}+\dfrac{2}{...

I have just learnt implicit differentiation and am trying to understand it using the chain rule to create deeper understand/link to previously-taught concept instead of just understanding it in ...

Good morning. I am trying to find the arc length of the cardioid $r=2\sin{\theta}-2$. After plugging in $r$ and $\frac{dr}{d\theta}$ into the arc length formula we get
$$s=\int_\alpha^\beta\sqrt{(2\...

For an integral $\int^b_af(x)dx$,it i substitute $x$ with $at$, the I have ${1\over a}\int^{b\over a}_1 f(t)dt.$
What confuses me is that why would we have a $1\over a$ front. From the integral's ...

The Theorem of Integral by Substitution states that:
Let $I \subseteq \mathbb{R}$ interval and $\phi:[a,b] \to I$ a
differentiable function with integrable derivative. Suppose $f:I \to
\mathbb{R}...

I have an exercise in mutivariable calculus where I am to transform a function $u(x,y) = u(\rho, \varphi)$ from and to polar coordinates by expressing
$\partial u/\partial \rho$ and
$\partial u/\...

I encountered this problem in class, and I did not understand how to solve it:
The problem:
Find all functions $ f \in CΒ²$ such that
$$
\dfrac{\partialΒ²f}{\partial xΒ²} + \dfrac{\partialΒ²f}{\partial ...

Admit that $f$ has a second derivative find the integer $m$.
$$m\int_{0}^{1}xf''(2x)dx = \int_0^2xf''(x)dx$$
So I took $2x=u$ where $du/dx=2$ and I plugged in the integral getting
$$\frac{m}{4}\...

When trying to solve $\int\frac{dx}{\sqrt{x^2-a^2}}$ we can substitue $x$ in two ways:
METHOD 1: $x=a\sec(\theta)$ and $dx=a\sec(\theta)\tan(\theta) d\theta$ and $\theta=\sec^{-1}(\frac{x}{a})$.
...

$$I=\int\frac{1}{2+\cos\theta}d\theta$$
When trying to solve this integral my calculus-book states that we can use the following substitution:
$$x=\tan(\theta/2)$$$$\cos\theta=\frac{1-x^2}{1+x^2}$$$$...

the integral I am solving is:
$$I=\iint_Sxy\sqrt{1+x^2+y^2}\,dx\,dy$$
where $S$ is a rectangle $0\le x\le1, 0\le y\le1$
after $$x=\rho\cos{\varphi},y=\rho\sin{\varphi}$$
$$dy\,dx=\rho \,d\rho \,d\...

I am in the process of finding the volume of the solid $E$, where
$$E = \{(x, y, z) \in \mathbb{R}^{3}: x > 0, y > 0, \sqrt{x} + \sqrt{y} \leq 1, 0 \leq z \leq \sqrt{xy} \}.$$
Now, this is an ...

If $a$, $b$, $c$ are real numbers, I have to prove:
$$a^4+b^4+c^4+(\sqrt {3}-1)(a^2 b c+a b^2 c+a b c^2 )\ge \sqrt {3} (a^3 b+b^3 c+c^3 a)$$
Since $$a^4+b^4+c^4 \ge abc(a+b+c)$$
then it is enough ...

- integration
- calculus
- inequality
- trigonometry
- indefinite-integrals
- definite-integrals
- polynomials
- algebra-precalculus
- fractions
- radicals
- differential-equations
- uvw
- a.m.-g.m.-inequality
- multivariable-calculus
- cauchy-schwarz-inequality
- contest-math
- systems-of-equations
- real-analysis
- trigonometric-integrals
- limits
- quadratics
- derivatives
- logarithms
- functions
- logic

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