Substitution Definition and 816 Threads

Homework Statement Find the integral \int \frac{3x+1}{(x^2-x-6)\sqrt{3x^2+4x-7}}\mathrm dx 2. The attempt at a solution I have tried the types of substitutions of irrational functions, and Euler substitutions. However, it seems that nothing simplifies this integral. What substitution is...

Homework Statement Find the integral \int \frac{2\sqrt[5]{2x-3}-1}{(2x-3)\sqrt[5]{2x-3}+\sqrt[5]{2x-3}}\mathrm dx 2. The attempt at a solution...

bapowell submitted a new PF Insights post The Monographic Substitution Cipher: From Julius Caesar to the KGB Continue reading the Original PF Insights Post.

Homework Statement The problem is the integral attached Homework Equations sec2(u)=(1+tan2(x)) a2+b2=c2 ∫cos(u)=-sin(u)+C The Attempt at a Solution The solution is attached. I am wondering if someone could give me a hint where I went drastically wrong or where I possibly dropped a negative...

Homework Statement Im looking over the notes in my lecture and the prof wrote, \int_{0}^{2} \pi(4x^2-x^4)dx=\frac{64\pi}{15} Im wondering what's the indefinite integral of this equation. Homework Equations using u substitution The Attempt at a Solution \int \pi(4x^2-x^4)dx= \pi \int...

Hello. I'm new to this forum. I'm starting a PhD – it's going to be a big long journey through the jungle that is CFD. I would like to arm myself with some tools before entering. The machete is Cartesian Tensors. I know the rules regarding free suffix's and dummy suffixes, but I'm having...

W.8.1.18 $$\int_{-1}^{1} \left(x^4-2x\right)^6\left(2x^3-1\right)\,dx $$ $u=x^4-2x$ $du=4x^3-2 dx =2(2 x^3-1) dx$ $$\frac{1}{2}\int_{-1}^{1} u^6\,du $$ Is this ok

I was wondering if you could do a trig substitution with cosine instead of sine. All the textbooks I have referred to use a sine substitution and leave no mention as to why cosine substitution was not used. It seemed that it should work just the same, until I tried it for the following Fint...

Homework Statement The compounds P, Q and S, were separately subjected to nitration using HNO3/H2SO4mixture. The major product formed in each case respectively, is : Homework EquationsThe Attempt at a Solution This is a standard EAS reaction. I know the answer will be either C or D, because...

Homework Statement Show by appropiate substitutions that ∫ (e2z-1)-0.5 dz from 0 to infinity is equivalent to ∫(1-x2)-0.5 dx from 0 to 1. Thus, show that the answer is π/2. Homework EquationsThe Attempt at a Solution Where to begin! I tried the substitution e2z= 2-x2, but this then transforms...

Homework Statement Let: ##I=\int _{-1} ^{1}{\frac{dx}{\sqrt{1+x}+\sqrt{1-x}+2}}## Show that ##I=\int_{0}^{\frac{ \pi}{8}}{\frac{2cos4t}{cos^{2}t}}## using ##x=sin4t##. Hence show that ##I=2\sqrt{2}-1- \pi## Homework EquationsThe Attempt at a Solution The substitution is ##x=sin4t## which...

When using trigonometric substitution in calculus you're supposed to always keep in mind the domain of the angle. In the case of √(x2-a2) (where "a" is a number >0) you use x=a⋅arcsec Θ for the substitution. For trigonometric substitution, textbooks state that the domain of Θ must be...

Hello, In finding a taylor series of a function using substitution, is it possible to use substitution for known taylor series of a function ,using different centers, and still get the same result. For example, if we have the function 1/(1+(x^2)/6) is it possible to use the taylor series of...

Homework Statement By making the transformation u= x^αy where α is a constant to be found, find the general solution of[/B] y'' + (2/x)y' + 9y=0The Attempt at a Solution I've worked out y,y',y'' and subbed them into get x^-au'' + x^a-1(2-2a)u' + x^-a-2(x^2-a(a-1))u =0...

CH3Cl +AgF :CH3F +AgCl Guys if I'm correct this is a nucleophilic substitution reaction and here I'm not getting the reaction because from my view chlorine atom is more nucleophilic than fluorine atom and carbon atom is more electronegative than silver atom so how the reaction is taking place?

Homework Statement Using the substitution x = 2sinθ, show that \int \sqrt{4 - x^2} dx = Ax\sqrt{4 - x^2} + B ⋅ arcsin(\frac{x}{2}) + C whee A and B are constants whose values you are required to find. Homework EquationsThe Attempt at a Solution x = 2sinθ \frac{dx}{dθ} = 2cosθ dx = 2cosθ ⋅...

Homework Statement Evaluate the integral: integral of dx / (4+x^2)^2 Homework Equations x = a tan x theta a^2 + x^2 = a^2 sec^2 theta The Attempt at a Solution x = 2 tan theta dx = 2sec^2 theta tan theta = x/2 integral of dx / (4+x^2)^2 = 1/8 integral (sec^2 theta / sec^4 theta) d theta =...

Homework Statement Evaluate \int{\frac{x^2}{(1-x^2)^\frac{5}{2}}}dx via trigonometric substitution. You can do this via normal u-substitution but I'm unsure of how to evaluate via trigonometric substitution. Homework EquationsThe Attempt at a Solution Letting x=sinθ...

Homework Statement x(dy/dx) - y = sqrt(xy +x2)Homework EquationsThe Attempt at a Solution I got up to this point: u=y/x dy/dx = (sqrt(xy+x2))/x + y/x And then the solution shows this: dy/dx = y/x + (y/x+1)½ Please help, I don't understand how they got to that point.

Hi, I'm currently taking ap calc bc as a senior in high school. Since trig sub and power reduction formula is not apart of the ap curriculum our class is skipping it. Assuming I pass the test and get credit for it, I will probably skip calc 2 in college. If I continue to study math and physics...

Homework Statement Evaluate the Integral: ∫sin2x dx/(1+cos2x) Homework EquationsThe Attempt at a Solution I first broke the numerator up: ∫2sinxcosx dx /(1+cos2x) 2∫sinxcosx dx /(1+cos2x) Then I let u = cosx so that du = -sinx dx -2∫u du/(1+u2) And now I'm stuck. I thought about turning...

[Prefix] When we do trigonometric substitutions (such as for the integral x^3/(a^2-x^2)^2), we say something like "let x = asinp for -pi/2 <= p <= pi/2" then we carry on and solve the integral. However, sometimes our answer is ugly and we get some term in our expression like "cosp"- so we draw...

Homework Statement (x+ln(x+sqrt(x^2-1)))^3 / (sqrt(x^2-1)) The Attempt at a Solution I have tried so many different things with this integral but cannot seem to get anywhere with it. Its almost so nicely an inverse coshx but not quite. Any help?

I have been working on a math problem and I keep getting the some type of PDEs. x*dU/dx+y*dU/dy = 0 x*dU/dx+y*dU/dy+z*dU/dz = 0 ... x1*dU/dx1+x2*dU/dx2+x3*dU/dx3 + ... + xn*dU/dxn= 0 dU/dxi is the partial derivative with respect to the ith variable. Does anyone know about this type of PDE...

Hello, I found something surprising (at least to me) while looking at the following integral: \int \sqrt{\frac{e^x-1}{e^x+1}} dx Wolfram Alpha suggests the following substitution as the first step: u = \frac{1}{e^x+1} Which leads to the following integral: \int \frac{\sqrt{1-2u}}{(u-1)u} du The...

$\int\frac{3{x}^{3}}{\sqrt{4{x}^{2}-1}}dx $ I wasn't sure what substitution to use due to what is in radical? $x=\frac{1}{2}\sec^2 \left({\theta}\right)\ dx=\frac{\sin\left({\theta}\right)}{\cos^3\left({}\right)}$

Look, I was wondering if substituting the variable more than once is valid and hence the definite integral intervals change this way. Consider the following integral (I'm working for finding the volume of a solid of revolution): *\pi \int_{-3}^{5}3^{2}-(\sqrt{\frac{y+3}{2}}+1)^2dy Personally I...

Hello, I am trying to understand the resolution of the following KdV equation. I try to demonstrate it by myself. The solitary wave solution is : At first, I created new variable as follows so I could transform the PDE into an ODE. A = A(p) p = g(x,t) g(x,t) = x - ct I succeeded to...

$\int{x}^{2}\sqrt{1-{x}^{2}}dx$ $u=\sec\left({x}\right)\ du= \frac{\sin\left({x}\right)}{{\cos\left({x}\right)}^{2}}dx$ I pursued this but got lost, maybe I don't need a trig subst.

Find the marginal rate of technical substitution for the following production function: Q=10(0.2L^{-0.5} +0.8K^{-0.5})^{-2} Here is my attempt so far: \frac{\delta Q}{\delta L}=[10(-2)][0.2K^{-0.5}+0.8L^{-0.5})^{(-2-1)}*[0.8*(-0.5)]L^{(\frac{-1}{2}-1)}=[(-20)*(-0.4)](0.2K^{-0.5}...

There is an ever constant increase of CO2 and CH4 in the atmospheric, thanks to global warming and greenhouse gases. I came across this equation while looking over changing gas compositions in the atmosphere. CO2 + CH4 ----> 2H2O + C2 Can someone explain me how I can figure out the amount of...

Hi, I'm not sure if this should actually be in the "homework" section instead. I'm posting it here because it's more of a pedagogy question, I think, but I could be wrong about that also. Ok, I tutor calculus, and when I do u-substitution, I always solve for something (not always dx), so that...

Homework Statement $$\int\frac{x^2+3}{x^6(x^2+1)}dx$$ Homework Equations None The Attempt at a Solution I easily got the answer using partial fractions by splitting the integral as ##\frac{Ax+B}{x^2+1}+\frac{C}{x}+\frac{D}{x^2}+\frac{E}{x^3}+...+\frac{H}{x^6}## and then finding the...

Homework Statement Use the substitution ##x=X+h## and ##y=Y+k## to transform the equation ##\frac{dy}{dx}=\frac{2x+y-3}{x-2y+1}## to the homogenous equation ##\frac{dY}{dX}=\frac{2X+Y}{X-2Y}## Find h and k and then solve the given equation Homework EquationsThe Attempt at a Solution If I...

Homework Statement Evaluate the integral of (x+1)5^(x+1)^2 Homework EquationsThe Attempt at a Solution I set my u=(x+1) making du=1dx. This makes it u*5^u^2. I integrated the first u to be ((x+1)^2/2) however I don't know what to do with the 5^u^2

Let ##x'=1/u' \Rightarrow dx' = \frac{-1}{u'^2} du'##. Then the integral ##\int_{x_0}^{x} x' dx'## turns into ##\int_{1/u_0}^{1/u} \frac{-1}{u'^3} du'##. Here comes the fallacy: ##\int_{1/u_0}^{1/u} \frac{-1}{u'^3} du' = [\frac{1}{2} \frac{1}{u'^2}]_{1/u_0}^{1/u} = \frac{1}{2} (u^2-u_0^2)##...

Hey guys I can't solve one integral ∫1/√(1+cosx) dx I have started like ∫1/√(1+(cos^2 x/2 -sin^2 x/2)) dx = ∫1/√(cos^2 x/2 + cos^2 x/2) dx = ∫1/√(2cos^2 x/2) dx = 1/√2 ∫1/(cos x/2) dx = { substitution t= x/2 dx= 2dt } = 2/√2 ∫ 1/cost dt= 2/√2 ∫1/ ( cos^2 t/2 - sin^2 t/2) dt = 2/√2 ∫1/(cos^2...

Homework Statement Integral of $$ x^3\sqrt{x^2+16}dx $$ answer should give $$ 1/5(x^2+16)^{5/2} -16/3(x^2+16)^{1/2}+C $$ Homework Equations x=atanθ The Attempt at a Solution Mod note: The integral is ##\int x^3 \sqrt{x^2 + 16} dx## The published answer is ##1/5(x^2+16)^{5/2}...

Homework Statement I am trying to understand a substitution used to solve for the center of mass of a solid uniform hemisphere as in this post: https://www.physicsforums.com/threads/solid-hemisphere-center-of-mass-in-spherical-coordinates.650663/#post-4151797[1] I completely understand the...

So I just started my DE class and I'm kinda stuck on solutions by substitutions. My book explains it as just having a homogeneous function of degree α, we can also write M(x,y) = xαM(1,u) and N(x,y) = xα (1,u) where u = y/x I don't understand how the substitution simplifies our life ( there's...

Recently I started seeing integral calculus and right now we are covering the topic of the antiderivative. At first sign it was not very difficult, until we started seeing integral variable substitution. The problem starts right here: Let's suppose that we have a function like this: \int...

Homework Statement Evaluate the following integral using a change of variables: \int\frac{dx}{\sqrt{1-\sin^4{x}}} Homework Equations If f(x)=g(u(x))u'(x) and \int g(x)dx = G(x) +C then \int f(x)dx = G(u(x))+C The Attempt at a Solution It seems helpful to first simplify a little to obtain...

$\int\frac{x}{9+x^4}dx$ $$u=x^2\ du=2x\ dx\ \ x=\sqrt{x}$$ I assume this going to have a trig answer but I didn't know how to deal with the $$dx$$

I was wondering if there is a convenient way of checking if a substitution is correct or not. For example, I tried solving for ∫(1/(a^2-x^2)dx using two different substitutions, x=acosu and x=asinu giving different solutions. I got the integral as arcsin(x/a) using x=asinu and -arccos(x/a) using...

I was reading in this book: Supergravity for Daniel Freedman and was checking the part that has to do with Extremal Reissner Nordstrom Black Hole. He was using killing spinors (that I am very new to). I was understanding the theory until he stated with the calculations: He said that the...

It is common that we replace \int u(x)v'(x)dx by \int udv where both u and v are continuous functions of x. My question is, must we ensure that u can be written as a function of v before applying this? The above substitution method is involved in the proof of integration by parts but I cannot...

Homework Statement 8t^2 * y'' + (y')^3 = 8ty' , t > 0 Homework EquationsThe Attempt at a Solution I tried using the substitution v = y' to get: 8t^2 * v' + v^3 = 8tv I rewrote it in the form 8t^2 * dv/dt + v^3 = 8tv, and then moved the v^3 to the other side to get 8t^2 * dv/dt = 8tv - v^3...

I don't understand the question: "First make a substitution and then use integration by parts to evaluate the integral" \int sin \sqrt{x} dx What does it have in mind by "substitution"?

Does anyone know of a derivation or justification of Euler's substitution formulas for evaluating irrational expressions? In other words, to evaluate integrals of the form: \int R(x,\sqrt{ax^2+bx+c}) You can use Euler's substitutions: 1. \sqrt{ax^2+bx+c} = t \pm \sqrt{a}x, a>0 2...

I just have a few questions. When using a trig substitution does it have to be under a radical ? eg, suppose I wanted to integrate (x2)/(x2-9), I used a trig substitution of x = 3sec(t) and got the wrong answer and so apparently I had to use partial fractions