Questions tagged [substitution]

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

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How to transform triple integral $\iiint_\Omega \sqrt{1- \frac{x^2}{a^2}- \frac{y^2}{b^2} - \frac{z^2}{c^2} }\ dx dy dz$

I have stumbled across this triple integral $$\iiint_\Omega \sqrt{1- \frac{x^2}{a^2}- \frac{y^2}{b^2} - \frac{z^2}{c^2} }\ dx dy dz$$ where $$\Omega =\left\{(x,y,z)\in{\cal{R}}^3\ \bigg| \ \frac{x^2}{...
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5 votes
3 answers
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What did I do wrong in solving $\int\sec^{-1} x\,{dx}$?

I used integration by parts: let u=$\sec^{-1}\,x$, dv=dx, then du=$\frac{1}{|x|\sqrt{x^2-1}}$, v=x. I = $x\sec^{-1}\,x\;-\;\int\frac{x}{|x|\sqrt{x^2-1}}dx\\$ Integration of $\int\frac{x}{|x|\sqrt{x^2-...
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2 answers
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Evaluate the algebraic indefinite integral $\int\frac{\sqrt{22+10x+x^2}}{x}dx$ [duplicate]

$$\int \frac{\sqrt{22+10x+x^2}}{x}dx$$ This is what I tried $$\int\frac{\sqrt{(x+5)^2-3}}{x}dx$$ I substituted $x+5=\sqrt{3}\sec\theta$ then got this $$\int\frac{3\tan^2\theta\sec\theta}{\sqrt{3}\sec\...
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4 votes
1 answer
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Show equality between these two series

I have a simple (?) index substitution (?) problem, where I want to show that $$ \sum_{n=0}^\infty \sum_{m=0}^n a_n b_m c_{n-m} = \sum_{j=0}^\infty \sum_{k=0}^\infty a_{j+k} b_k c_j $$ but whatever I ...
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Problems with integral substitution that involves weird physics index notation and distributions.

I‘m supposed to show that the 3x3 matrix T, whose components are given by: $$ T^{ij}=\int_{\partial B_1(0)} \frac{x^i x^j}{|x||x|}d \Omega $$ ($x^i$ the i’th component of $x$, $\partial B_1(0)$ the ...
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  • 355
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Difficulty understanding the substitution $dS(y)=rdS(z)$

In the proof for the mean value property for harmonic functions we start with the integral $\frac{1}{2\pi r}\int_{\partial B(x,r)}u(y)dS(y)$ where $\partial B(x,r)$ is the boundary of the disk $B(x,r)$...
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Substitution in 2 variable limits

Goodmoring, I'm having difficulty in resolving 2 variable limits with some variable substitution. I can't understand when the substitution is legit or not. My calculus teacher told me that I've to ...
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Convergence of improper integral, $\cos(1/x)$

I'm trying to deduce weather this improper integral is convergent or not: $$ \int_{0}^{1}\dfrac{\cos(\frac{1}{x})}{x}dx. $$ I've tried using Dirichlet's test for convergence, yet I cant seem to ...
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Converse of renaming substitutions

In the paper Theory Unification in Abstract Clause Graphs, Hans Jorgen Ohlbach states that, given the renaming substitution $\sigma$, the converse substitution $\sigma^{c}$ is defined as $x=\sigma(y) \...
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Area under curve in two different ways

I am currently trying to verify the computation of following integral using two different substitutions. The area below $f$ on the unit the circle with $$f(x,y) = \frac{x^2 + y^2}{4} + \frac{xy}{2} $$ ...
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Does Euler Third Substitution gives different answers?

In the book "Special Techniques for Solving Integrals: Examples and Problems" by Khristo N. Boyadzhiev, I am given the integral $$F(x)= \int \frac{dx}{(x+3)\sqrt{3x-x^2 -2}}$$ as an example. ...
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Fredholm Integral substitution

This is the task Show that the integral equation $$ f(x)=\int_{0}^{1}|x-y|^{\alpha} f(y) d y+B(x) $$ has a unique solution $f \in C[0,1]$ for all $\alpha>0$ and $B \in C[0,1]$. I do not ...
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U substitution to turn a double integral into a convolution?

I'm trying to work through the results of a classic paper in super-resolution microscopy (C. Sheppard, Optik, 1988). The main result sort of hinges on writing the double integral: $$ \int dx \int dx^\...
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3 votes
1 answer
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I have the integral $\int_{-1}^{1} \frac{\arccos(x)}{1+x^2} \,dx $ and some questions. Any help appreciated!

$$\int_{-1}^{1} \frac{\arccos(x)}{1+x^2} \,dx $$ Hi everyone! Sorry for my poor formatting skills, I'm still quite new to this platform. I do not know how to solve this integral. Things that I tried ...
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Help evaluating a limit.

Using the substitution $u=\frac{1}{x}$, show that $\int_{\frac{1}{2}}^{2} \frac{\ln{x}}{1+x^2} \,dx = 0$. Hence or otherwise, evaluate the limit $\lim_{n\to\infty} \sum_{k=1}^{3n} \frac{1}{2n} \times \...
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1 vote
1 answer
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How and why does this variable substitution in this integral work?

$$\int_{-5}^5 \frac{1}{1+2^{\arctan(x)}} dx$$ In the solution of this example authors do such thing: $$\int_{-5}^5 \frac{1}{1+2^{\arctan(x)}} dx = [x = -t; dx = -dt; t = -x] = \int_5^{-5} \frac{-1}{...
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2 votes
2 answers
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Evaluating $\int_{1/\sqrt{2}}^1\frac{1}{(3y^2-1)\sqrt{2y^2-1}}\,\mathrm{d}y$; how do you avoid using a complex substitution?

$\newcommand{\d}{\mathrm{d}}$The given exam question - I provide the beginning for context: Let: $$I=\int\frac{1}{(b^2-y^2)\sqrt{c^2-y^2}}\d y$$Where $b,c\gt0$, and employ the substitution $y=\frac{...
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Effective method to solve $\frac{\sqrt[3]{1+x} -\sqrt[3]{1-x}}{\sqrt[3]{1+x} +\sqrt[3]{1-x}} = \frac{x(x^2+3)}{3x^2+1} $

I want to know, is there an easy method to solve below equation $$\frac{\sqrt[3]{1+x} -\sqrt[3]{1-x}}{\sqrt[3]{1+x} +\sqrt[3]{1-x}} = \frac{x(x^2+3)}{3x^2+1} $$ I tried it by plotting and find the ...
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3 votes
2 answers
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Integration by substitution: use of appropriate substitutions with roots of even index

I have the following integrals $$\int x\sqrt{x-1}\, dx, \quad \text{and} \quad \int \sqrt{e^x-1}\,dx$$ I am not interested in solving integrals but rather in a consideration that I have thought when I ...
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What is the idea behind Subsitution? [closed]

What is the Logical formulation of Substitution? Specifically how do we treat looking at a formula and think about numerical values, or truth values. What is the idea behind our treatment of ...
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1 vote
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Limits of integral over a region after substitution

I have the integral of a function $f(x,y,z)$ defined over the region $a<\sqrt{x^2-y^2-z^2}<b$; that is, $$I\equiv \int_{a<\sqrt{x^2-y^2-z^2}<b}dxdydzf(x,y,z).$$ I realized that defining ...
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How to integrate $\frac{\cos2x}{ \sin x + \sin 3x}$?

I thought of using trig identities to get rid of $\cos2x$ and $\sin3x$, then use Weierstrass substitution, but I got myself into big trouble as the expression got too complicated. Is there a simpler ...
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integral after substituion of $x\to-y$

I just cant spot the mistake in my calculation. As I said in the title, I used the substituion $x\rightarrow-y$, $dx\rightarrow-dy$ and $a<b$ \begin{align} F(b)-F(a)&=\big[F(x)\big]_a^b=\int_a^...
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1 vote
1 answer
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Integration by substitution, $\int_1^a f(x^s) dx = \int_1^{a^s} f(x) \dfrac{1}{sx^{1-1/s}} dx$

I am trying to show that, for a function $f: \mathbb{R} \to \mathbb{R}$, and any $s>0$, $a>1$, we have that $$ \int_1^a f(x^s) dx = \int_1^{a^s} f(x) \dfrac{1}{sx^{1-1/s}} dx.$$ I am trying to ...
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Find $C$ such that $\sum_{i=1}^{n}\frac{x_i^3}{\alpha x_i^{2}+\beta x_{i+1}^{2}}\geq \frac{\sum_{i=1}^{n}x_i}{C}$ is true

Problem Let $x_i>0$ and $n\geq 3$ then find the best constant which is a natural number $C=\alpha+\beta$ with $\alpha,\beta >0$ natural numbers such that $$\sum_{i=1}^{n}\frac{x_i^3}{\alpha x_i^...
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Using substitution method solve the recurrence $T(n) = 3T(\frac{n}{3}) + \frac{n}{(\log n)}$

I try to do it as seen in this answer but since my log is base 3 it doesn´t turn into i image of my incomplete attempt to find complexity if my n=3^k then my sum´s division is 0, so I must be doing ...
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2 answers
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How do I solve the integral $\int\frac{dz}{z\sqrt{4z+z^2}}$? [closed]

The proposed integral is given by $$\int \frac{dz}{z\sqrt{4z+z^2}}$$ I'm trying to answer this equation using the Reciprocal Substitution method, how should I approach this? Should I rewrite or ...
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Determining the value of $f(x)$ if we substituted the value of $a$.

If given a variable $y$ such that $y=f(x)$ can we talk about the value of $y$ for a (separate) number $a$ if we were to substitute the value of $a$ for $x$ in the expression defined as $f(x)$? Case 1: ...
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3 votes
1 answer
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Is this a viable technique for integration that I should persue exploring? Thoughts?

I started with the integral of: $$\int \frac{\ln(x)}{x}dx = \frac{\ln^2(x)}{2} + C$$ Which was very easy to integrate. Then, I moved to a more difficult problem: $$\int \frac{\ln(x-t)}{x} dx$$ I ...
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1 vote
1 answer
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Evaluate indefinite integral using susbtitution

I have the following integral to evaluate. I think it should be done by substitution but I get stuck midway when I use $u=x^5$ and $du=5x^4dx$ $$\int x^{14}\sqrt{x^5+2}\,dx$$
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How to integrate this equation to get to the second line?

How does one go from this: $ \int \frac{l_0(1 − ξ/x_1)}{\sqrt{κ_0^2(1 − x_0/x_1)^2 − l_0^2(1 − ξ/x_1)^2}} dξ$ to this: $ \frac{x_1}{l_0}\sqrt {κ_0^2(1 − x_0/x_1)^2 − l_0^2(1 − ξ/x_1)^2}$ What i've ...
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Substitution of main function (y) into differential equation

I have a problem understanding how these substitutions are possible in examples (2) & (3), clearly this is not the main function as the place of the (A) & (B) parameters are switched in ...
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How to prove that $\int_{0}^{1} x^a(1-x)^bdx = \int_{0}^{1} x^b(1-x)^adx$?

So I'm giving this problem as an extra exercise: $$\int_{0}^{1} x^a(1-x)^bdx = \int_{0}^{1} x^b(1-x)^adx$$ What I tried is let $u = 1 - x$, $x = 1 - u$ and $du = -dx$ and then substituted in and got $$...
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4 votes
3 answers
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How can I make this 1st order ODE separable?

$$ y (2+3xy) \,{\rm d} x = x (2-3xy) \,{\rm d} y $$ I tried using the substitution $y=\frac{v}{x}$, but that didn't get me far. Then I tried using the substitution $y=vx$, but that didn't work either. ...
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1 vote
1 answer
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How can I proceed from here?

$$\frac{dy}{dx}=\frac{2x+3}{y+x-2}$$ Using the substitution: $y=vx$ I turned this expression into $$\frac{dv}{dx}\cdot x +v=\frac{2x+3}{vx+x-2}$$ Usually, at this stage, you can cancel a few things ...
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0 votes
1 answer
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Solving $\int_0^2 z^{n-a-2} (2-z^{-1})^b dz$ by using beta function

I'm wondering which substitution I need to transform the integral below like the integral in the definition of the beta function (I know that $a>-1$ and $b>-1$, and $n \in \mathbb{N}$): $$ \...
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0 answers
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Solving $\int_0^2 x^a (2-x)^b dx$ by using beta function [duplicate]

I'm wondering which substitution I need to transform the integral: $$ \int_0^2 x^a (2-x)^b dx $$ so that the endpoints will become $0$ and $1$ , and the second argument of the form $(1-x)^c$ in order ...
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0 answers
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Integral transformation for $f\in L^1([0,1]^n)$

For $f\in L^1([0,1]^n)$ and an inversible matrix $\Phi$ with $|\mathrm{det}(\Phi)|=|a|\in (0,1)$. I want to make sens of the right hand side of the following equation, which follows from the ...
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-1 votes
1 answer
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If $\int_{0}^{a}f(x)dx=I$, then $\int_{0}^{a}f(x)f(a-x)dx=\dots$

A well known result of definite integrals is: $$\int_{0}^{a} f(x)dx=\int_{0}^{a} f(a-x)dx$$ It is very easy to verify this result by substituting $x=a-y \implies dx=-dy$. When $x=0, y=a$, and when $x=...
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1 vote
1 answer
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Question about finding $du$ from $dx$ when integrating by substitution

Example 1: $$\int\frac{\tan^2(\ln x)}{x}dx$$ Method 1: $$\text{Let}\ u=\ln x$$ $$\implies du=d(\ln x)$$ $$\implies du=\ln'(x)dx$$ $$\implies du=\frac{1}{x}dx$$ Method 2: $$\text{Let}\ u=\ln x\tag{...
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2 votes
2 answers
98 views

How do I know which function to substitute the variable when solving problems by the "Integration by substitution" method?

In Calculus, we use the "Integration by Substitution method" to integrate variables that are otherwise difficult to do by the conventional method. What I don't understand which function do ...
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0 votes
2 answers
61 views

How does $u$-substitution work: $\int \frac{\sqrt y e^{-y/2}}{2} \mathrm{d}y$

I have been banging my head on the table for 3 hours trying to understand the final step here. Where did $-2^{(3/2)}$ come from? Why is there a $-u$ inside the square root?
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0 votes
2 answers
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Substituting $it+\ln t$ with $u$ in challenging integration procedure

Regarding this post I would like to ask the community for hints regarding this problem. Since the steps done on that original post, did not yield the correct result, I have done a different approach. ...
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2 votes
3 answers
105 views

Evaluating $\int\frac{\sec^2(x)}{(4+\tan^2(x))^2}\, dx$

How to solve this integral?$$\int\frac{\sec^2(x)}{(4+\tan^2(x))^2}\, dx$$ I've tried the following: Starting by substituting $\tan(x) = 2\tan(\theta)\implies \sec^2(x)\ dx= 2\sec^2(\theta)\ d\theta$ $$...
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  • 1,810
4 votes
3 answers
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Why is this secant substitution allowed?

On Paul's Math Notes covering Trig Substitutions for Integrals we start with an integral: $$\int{{\frac{{\sqrt {25{x^2} - 4} }}{x}\,dx}}$$ Right away he says to substitute $x=\frac{2}{5}\sec(θ)$. Why ...
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2 votes
2 answers
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is this definite integral correct?

I have this definite integral $$ \int^{2}_{1} \frac{e^{1/x}}{x^4}dx$$ this is my attempt: I used u-substitution. $u = 1/x$, and then $-du = 1/x^2 dx$ I rewrote $1/x^4$ as $(1/x^2) * (1/x^2)$ now, I ...
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1 vote
2 answers
55 views

Validity of trigonometric substitutions

Let us look at a term $\dfrac{x}{\sqrt{1+x^2}}$. Here $x>0$. Now we can make a trigonometric substitution $x=\tan A$. But why does this $A$ have to be in $(0,\frac{\pi}{2})$? I don't understand ...
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0 votes
1 answer
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Integrating $\frac{0.36h^2 + 1.44h + 1.44}{0.034 - 0.012136 \sqrt{h}}$ - Stuck with one part

I have been trying to follow how to integrate $\frac{0.36h^2 + 1.44h + 1.44}{0.034 - 0.012136 \sqrt{h}}$ using https://www.integral-calculator.com/ , but when it gets to $(2 \div 5295931061521 \times ...
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2 votes
1 answer
58 views

(Integration) Question on U-Substitution Mistake

For context, I was playing with this integral recreationally: $$\int{20\sin(\frac{x^2}{35})}dx$$ I decided to try u-substitution, and got the following: $$u=\frac{x^2}{35}\space,\space du=\frac{2x}{35}...
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  • 81
4 votes
1 answer
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Quadratic-trigonometric integral

Problem I need to compute the following interval \begin{equation*}\int_{t_\text{s}}^{t_\text{e}} \cos\left(a+b\tau+c\tau^2\right)\text{ d}\tau\end{equation*} where $t_{\text{s}},t_{\text{e}},a,b,c$ ...
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