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Questions tagged [substitution]

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

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Double integral $ \iint_D (x^4-y^4) dx\,dy$

I have troubles with the following integral $$ \iint_D (x^4-y^4) dx\,dy $$ over D: $1<x^2-y^2<4, \sqrt{17}<x^2+y^2<5, x<0, y>0$ This is the same problem as in Compute $\iint_D (x^4-y^...
TerribleStudent's user avatar
2 votes
1 answer
78 views

Double integral of $x^2y+y \sin(x^9)$ dxdy

I have some troubles with the following double integral (in particular the part with sinus) $$ \iint_{D}\left[x^{2}y + y\sin\left(x^{9}\right)\right]{\rm d}x\,{\rm d}y\quad \mbox{where}\quad D\ \mbox{...
TerribleStudent's user avatar
0 votes
1 answer
122 views

Double integral of $xe^{-(x^2+y^2)}$

I have some troubles with the following double integral where D is $|x|\leq 1, |y|\leq 1$ $$ \iint_{D} xe^{-(x^2+y^2)} \,dx\,dy $$ I transform it to polar coordinates where $\theta~is [0,\pi /2]:$ $$ \...
TerribleStudent's user avatar
2 votes
0 answers
27 views

Simplifying the inverse normal cdf of an affine transformation of the normal cdf

I want to simplify expressions of the form $\Phi^{-1}(a*\Phi(x)+b))$. A general result would be nice, but if it's helpful, I'm particularly interested in two cases: $0<a<1$, $b=0$. $a=(1-b)$, $...
Leland Stirner's user avatar
7 votes
6 answers
692 views

The method of substitution in the problem of finding the integral

My teacher gave me a simple problem: Find $\int \dfrac{x}{\sqrt{x+1}} \, dx$. This is how I approached it: I set $u = \sqrt{x+1}$, which implies $u^2 = x + 1$, thus $2u \, du = dx$. Therefore, $$ \int ...
Math_fun2006's user avatar
0 votes
2 answers
158 views

Calculate quickly $\int\sqrt{a^2+b^2x^2}\ dx$ with $a,b>0$.

We have $$\int\sqrt{a^2+b^2x^2}\ dx$$ with $a,b\in \Bbb R $ and $>0$. To solve this integral I can use a hyperbolic substitution as $$x = \color{red}{ \frac{a}{b}\ \sinh u} $$ Consequently I will ...
Sebastiano's user avatar
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1 vote
1 answer
72 views

Integral of $\tan(A+B)\tan(A-B)$

Evaluate $\int \tan(2x+a)\tan(2x-a) dx$ I actually got to this integral while trying to solve for the actual function $\frac{1}{\cos(2x+a)(\cos(2x-a))}$. I multiplied and divided by $\cos 2a=\cos((2x+...
a_i_r's user avatar
  • 689
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How can I evaluate $ \int_0^\infty \frac{\ln(2e^x-1)}{e^x-1} dx $ using Feynman's trick? [duplicate]

I would like to find out a solution using just Feynman's method and Calculus I basic methods, to share with a novice student. I have found at Quora and this site some solutions containing ...
Arthur's user avatar
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1 answer
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How to solve the integral $\iint_D(x^2+y^2)^{-2} dxdy$ with where $D$ is $x^2+y^2\leq2, x\geq 1$ and after that the same integral but with $x\leq1$. [closed]

So my main question is after I use the Jacobian how do I write the new space $D^*$ when I substitute $x=r\cos θ$ and $y=r\sin θ$. I know that I will find what the range of $θ$ is by simply finding the ...
A Math Wonderer's user avatar
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0 answers
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How to solve the $\iint_D \frac{(x-y)^2}{1+x+y}dxdy$ where D is the trapezoid with edges $(1,0),(0,1),(2,0),(0,2)$ and using $u=1+x+y$ and $v=x-y$

I know that you need to find the Jacobian of $u,v$ and multiply it with the existing $f(x,y)$ inside the integral. But how can I then find the limits of integration for $u$ and $v$ after that since ...
A Math Wonderer's user avatar
2 votes
0 answers
33 views

Rewriting the second derivative of a function by substitution

I would like to know if the equation $$ \frac{d^2T(x)}{dx^2} = \frac{1}{2}\cdot\frac{d}{dT}\left(\frac{d}{dx}T(x)\right)^2\quad(1) $$ is true for a general function T(x). The function T(x) describes ...
Emann's user avatar
  • 21
1 vote
1 answer
165 views

When should I use the substitution $u=\frac{1-x}{x+1}$?

I see people use the substitution $u= \frac{1-x}{x+1}$ in integrals but I have no idea when to use it. I can see that this function $f(x)=\frac{1-x}{x+1}$ has an interesting property when trying to ...
pie's user avatar
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2 votes
6 answers
691 views

Help with a Calculus exercise (indefinite integral)

I just started Calculus II, and up until now, I only did concrete examples, like integrals of actual functions, for example, $\frac{1}{\cos(x)}$. I was given the following exercise and am not sure how ...
natitati's user avatar
  • 409
0 votes
3 answers
116 views

Find $f(x)$ assuming that $f(\sin x)+f(\cos x)=2x-\frac{\pi}{2}$

If $f(x)$ is a real valued function such that $$f(\sin x)+f(\cos x)=2x-\frac{\pi}{2}$$ Find $f(x)$. I did $x\to\arcsin x$ and then $x\to \arccos x$ and I obtained $2\arcsin x=2\arccos x$ or $x=\frac{...
MathStackexchangeIsMarvellous's user avatar
1 vote
5 answers
101 views

If $\log_7 5$ = a , $\log_5 3$ = b , $\log_3 2$ = c, then the logarithm of the number 70 to the base 225 is?

So, I've tried using the properties: $$\log_a b = \frac{\log_c b}{\log_c a}$$ and.. $$\log_a bc = \log_a b + \log_a c$$ And, the final simplification should be in the following options: $$A. \frac{1-a+...
Mune's user avatar
  • 13
1 vote
4 answers
140 views

Making the substitution $x=e^y$ in the integral $\int_0^1 x^{2n} \ln x/(1+x^2) dx$

$$ \mbox{The integral}\quad \int_{0}^{1}x^{2n}\frac{\ln\left(x\right)}{1 + x^{2}}{\rm d}x\quad \mbox{converges to}\quad r_{n} + {\rm C}s_{n} $$ when $n$ is a non negative integer. Where $r_{n},s_{n}$ ...
Pinteco's user avatar
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1 vote
2 answers
133 views

How do we solve the ODE $(2y+x)y' + 2 = y^2 + xy - y$ according to the substitution $v = y^{2} + yx$?

I am attempting to perform the following substitution in the non-linear ODE below: $$ v=y^2+yx $$ $$ (2y+x)y'+2=y^2+xy-y $$ After attempting a multitude of different approaches, I still am unable to ...
Crusader's user avatar
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0 answers
36 views

Integration measure for a strange substitution

I have a 2D integral over a momentum vector, i.e. $\int dp_x dp_y$ and the substitution for this is given by $$ \xi = |\vec{p}| + |\vec{p} + \vec{q}| , \, \, \, \eta = |\vec{p}| - |\vec{p} + \vec{q}|$$...
Johnny_T's user avatar
1 vote
0 answers
90 views

How to calculate $\int_{0}^{2\pi}(a+\sin{x})^{-3/2}dx$ [closed]

I wonder if there is an analytic solution for the following equation: $$\int_{0}^{2\pi}(a+\sin{x})^{-3/2}dx$$ Here, $a$ is a constant. Would you please give an advice?
donggun's user avatar
  • 37
1 vote
3 answers
152 views

$\int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr$

I'm trying to solve problem 3.3 from Jackson's Classical Electrodynamics, but I'm encountering some troubles solving $$ \int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr, \qquad l = 0,2,4,6,\ldots $$ ...
Peluche's user avatar
  • 135
5 votes
2 answers
103 views

Solve $\Bigl(2 \sin(x)D^2+2 (\cos(x)+\sin(x))D\Bigr)y+2y\cos(x)=\cos(x)$

How to solve the following differential equation? $$2 \sin(x)\frac{d^2 y}{dx^2}+2 \cos(x) \frac{dy}{dx}+2 \sin(x) \frac{dy}{dx}+2y \cos(x)=\cos(x)$$ This was given to me by one of my friends as a ...
Nairit Sahoo's user avatar
0 votes
2 answers
74 views

Problem with simplification of this indefinite integral

Let's consider this fairly easy problem. $$\int \sin x \cos x dx$$ which can be transformed into $$\dfrac12 \int \sin 2x \ dx$$ Setting $t = 2x$, we have $dt/dx = 2$ Substituting this value into our ...
Ddev's user avatar
  • 1
3 votes
1 answer
77 views

differential equations, substitution suggested by the equation

I took my examination in differential equations earlier; i would've gotten a perfect score, but i tripped in this problem: $$(1+5y\sin x)dy + y^4\cos xdx = 0$$ I used substitution suggested by the ...
lawrencium21's user avatar
0 votes
1 answer
24 views

Length of a regular arc

I have to prove that if two arc are (positive) equivalent, they have the same length. Let $\alpha:[a,b]\to \mathbb{R}^n$ and $\beta:[c,d]\to \mathbb{R}^n$ two differentiable arcs and let $h:[a,b]\to [...
Sigma Algebra's user avatar
0 votes
1 answer
72 views

Help on transformation of boundary conditions

I was working the transformation in this paper A new algorithm for solving classical Blasius equation by Lei Wang The boundary value problem is He used the transformations $$y=f''(\eta),x=f'(\eta)$$ ...
Mohamed Mostafa's user avatar
0 votes
0 answers
30 views

If I use this proposition to get an elementary primitive of $f$, I think $g(t)$ must be a monotonic function. ("Calculus" by Takeshi Saito)

The following is from "Calculus" (in Japanese) by Takeshi Saito. Proposition 4.3.9 Let $f(x)$ be a continuous function defined on an open interval $(a,b)$ and $g(t)$ be a continuously ...
佐武五郎's user avatar
  • 1,200
2 votes
2 answers
182 views

Just a simple algebra question

So I have this question: Solve: x = 3y, (a) x + y + z = 56, (b) x - 2y - 3z = -25 So you can substitute 3y for x to then eliminate the z by multiplying (a) by -3z then solve accordingly and you get z =...
Hogarth's user avatar
  • 21
1 vote
0 answers
35 views

definite integral involving u-sub with parameterized curves

While proving a pretty simple argument about reverse parametrization of a complex-valued function, I have an integral of the form: $$ \int_a^b f\bigl(\gamma(a+b-t)\bigr)\cdot\bigl(-\gamma'(a+b-t)\bigr)...
giorgio's user avatar
  • 583
2 votes
2 answers
77 views

Evaluate $\int{\frac{1}{x^3}}$ using u-sub.

I tried solving $\int{\frac{1}{x^3}}$ using u-sub instead of power rule and I got $-\frac{1}{2}x^{-2}$ instead of $\frac{x^4}{4}$. Its very possible I've made a very simple mistake or their is ...
Robert Barnett's user avatar
0 votes
2 answers
51 views

How should these types of integrals be solved? $\int\frac{1}{y}\frac{1}{\ln y}$

$$\int\frac{1}{y}\frac{1}{\ln y}dy$$ $$\int\frac{1}{y}\frac{1}{\ln^2y}dy$$ I know that i should use some substitution, but i don't understand how.Is there any other way than substitution? I've tried ...
GooDinosaur's user avatar
0 votes
0 answers
51 views

Python sympy: Expressions with an undefined function whose value at zero is known

I'm trying to evaluate an expression in sympy using a function $f(x)$. The function is unknown, but its value at $0$ is known $f(0)=0$. If it's any help, specifically I want to evaluate the Taylor ...
Amir Ban's user avatar
4 votes
0 answers
75 views

Trigonometric Substitutions related to Elliptic Integrals

Recently while studying Elliptic Integrals and related topics I have come across various Interesting Trigonometric Substitutions, examples given below 1. $$\int_0^{\pi/2}\frac{1}{\sqrt{1-x^2(\sin t)^4}...
Miracle Invoker's user avatar
1 vote
1 answer
65 views

Textbook says to integrate a fraction using 'Taylor's formula'?

I don't understand the solution my textbook gives for this problem: $$ \int \! \frac{x^3}{(x+1)^5} \, \mathrm{d}x $$ I thought it had to be done with partial fractions, but I couldn't get it right, ...
user avatar
0 votes
1 answer
55 views

What is wrong with substituting a constant in a system of equations?

Imagine that I have the following system of equations: \begin{equation} \left\{ \begin{aligned} x + 2 &= 3 \\ y - 5 &= 3 \end{aligned} \right. \end{equation} The solution (x=1, y=8) of this ...
Антон Бугаев's user avatar
2 votes
0 answers
86 views

Approach to evaluate the integral $\int \frac{1}{{x^3}\sqrt{x^2-a^2}}dx$ using basic substitution.

So I was attempting a basic indefinite integral problem which goes like $$\int \frac{1}{{x^3}\sqrt{x^2-a^2}}dx$$ for which I used the substitution $x=a\sec\theta$ which on differentiating gave $$dx=a\...
Aayush's user avatar
  • 59
0 votes
0 answers
75 views

How is this property of definite integral derived?

The property: $$ \int_a^b f(x) \, dx=\int_a^b f(a+b-x) \, dx $$ Derivation given in my textbook: Let $t = a+b-x$. Then $dt = -d x$. When $x=a, t=b$ and when $x=b, t=a$. Therefore, $$ \begin{aligned} \...
Nitish's user avatar
  • 33
0 votes
0 answers
24 views

Estimating an integral via substitution

I have a question about the computation of a certain integral. Let $\chi_{[0,B]}(\vert k-l\vert)$ be the characteristic function on the interval $[0,B]$. In order to estimate the following integral in ...
putti.123's user avatar
  • 339
0 votes
1 answer
37 views

What U substitution is being used here? (direct integral calculation of charge)

In my physics reading, we have the integral over $x_s$ from -L/2 to L/2 of $\frac{1}{((y^2+x_s^2)^{3/2})}$. Somehow, from this, the reading gets $\frac{x_s}{y^2*\sqrt{y^2+x_s^2}}$ from -L/2 to L/2. I'...
GalaxyMage13's user avatar
2 votes
1 answer
191 views

Why doesn't substiuting work here?

I know that:$$\int \cos x dx = \sin x +C$$ Substiute $x$ for $ax+b$: $$\int \cos(ax+b) dx = \sin(ax+b) +C$$ but according to my book: $$\int \cos(ax+b) dx = \frac{1}{a}\sin(ax+b) +C$$ Why doesn't ...
BadUsername's user avatar
6 votes
2 answers
226 views

Find all $x \in \mathbb{R}$ such that $1+\sqrt{1-\frac{1}{x}}+\sqrt{x-\frac{1}{x}}=x^2$.

This problem was posed by a friend of mine. He's got it from a social media post. We don't know how to solve it, but there are similar types of questions in this website, whose techniques I've tried ...
asd3weqsda zfewrewgsfd's user avatar
0 votes
4 answers
52 views

Substitution to get to a specific form

So I need to get the integral $$ \int_0^1 \frac{x+3}{x^2+x+2}dx $$ to the form $$ \int_a^b \frac{Au+B}{u^2+1}du $$ for some $a, b, A, B \in \mathbb R$. I can get either the numerator to the correct ...
user avatar
1 vote
2 answers
45 views

Problem on trigonometric substitution of an integral

Suppose we have to find the integral of the function $\sqrt{\frac{x}{1-x}}$. To solve this, I set $\sqrt{x} = -\sin\theta$, where theta belongs to [0, -π/2) which implies that $x = \sin^2\theta$. The ...
Rajesh Paul's user avatar
0 votes
0 answers
36 views

How to integrate a function with respect to the logarithm of a variable?

I am having trouble integrating the equation $\frac{dN}{dlogM} = C\left(\frac{M}{M_{br}}\right)^n$. I just need it between two limits, say $M_{l}$ and $M_{h}$. Sorry for a remedial question, it has ...
Samson A Johnson's user avatar
-1 votes
3 answers
103 views

Why does the antiderivative of $\int (1+u^2)𝑑u$ become $(u+ \frac{1}{3}u^3) + C$?

Example problem from Calc II lecture: $\int \sec^4(x)dx$ Worked out solution from lecture: $\int \sec^4(x)dx = \int (1+\tan^2(x))(\sec^2(x))dx = \int (1+u^2)du = $ Question: Why when doing the ...
user avatar
2 votes
4 answers
275 views

evaluate $\int \frac{x^3}{\sqrt{x^2+1}}dx$

Need help evaluating $\int \frac{x^3}{\sqrt{x^2+1}}dx$ I want to know why my working out is illegitimate. It looks illegal. Im just not sure by what mechanism it is illegal.
moon river's user avatar
1 vote
4 answers
129 views

Idea to find range of a function

I have to find range of this function $$ f(x) = \frac{x}{x^2+x+1} + \frac{x^2}{x^4+x^2+1} $$ I did it by using derivation as follows: $D_f(x)=\mathbb{R}$ so it suffices to check $\lim_{x \to \pm\infty ...
Khosrotash's user avatar
  • 25.2k
1 vote
1 answer
61 views

Definite integral substitution problem

I had to solve the following definite integral, i solved it by substitution but I checked many times and I can't find out why the result I get is wrong. I know it can be solved by parts, but I want to ...
NICOLA TROMBINI's user avatar
-1 votes
1 answer
39 views

How could I reason that $P=V^2 \div R$ because $P = iV$ and $V=iR$?

I'm trying to teach myself simple electronic formulas. I haven't done algebra in awhile, and I'm very rusty. I know that $P = iV$, and $V = iR$, therefore, $P = i^2R$. How could I figure out that $P = ...
LuminousNutria's user avatar
0 votes
0 answers
69 views

Generalizing an identity for polynomials [duplicate]

For any quadratic function, prove that: $$ f(x+3)-f(x) = 3[f(x+2)-f(x+1)] $$ This was a fairly straightforward problem, I solved it by assuming $f(x)=ax^2 + bx + c$ and by simplifying $g(x)=f(x+3)-3f(...
Cognoscenti's user avatar
2 votes
1 answer
64 views

Does Weierstrass substitution work for different arguments in the trig functions?

Suppose we have a two dimensional rational function $R(x,y)$. Then we can use Weierstrass substitution for following integral: $$\int R(\sin(x), \cos(x)) \, dx .$$ But can we use Weierstrass ...
haifisch123's user avatar

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