What is Integration by substitution: Definition and 84 Discussions
In calculus, integration by substitution, also known as usubstitution or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards".
Hi, PF, here goes an easy integral, meant to be an example of integration by parts.
Use integration by parts to evaluate
##\int \sin^{1}x \, dx##
Let ##U=\sin^{1}x,\quad{dV=dx}##
Then ##dU=dx/\sqrt{1x^2},\quad{V=x}##
##=x\sin^{1}x\int \frac{x}{\sqrt{1x^2} \, dx}##
Let ##u=1x^2##...
Is it possible to integrate the following function analytically?
##\int_{0}^{\infty} \frac{\exp{(\frac{A}{\tau}+B\tau+\frac{A}{\beta\tau})}}{\sqrt{\tau(\beta\tau)}}d\tau,##
where ##A##, ##B## and ##\beta## are real numbers. What sort of coordinate transformation makes the integral bounded...
I have seen the wikipedia's proof which can be found here: https://proofwiki.org/wiki/Integration_by_Substitution
However sometimes, we have problems where you have a ##d(x)## times ## f(g(x))## times g prime of x where we use substitution and it works but the proof didn't prove this...
Homework Statement
Question:
To solve the integral ##\int \frac{1}{\sqrt{x^24}} \,dx## on an interval ##I=(2,+\infty)##, can we use the substitution ##x=\operatorname {arcsint}##?
Explain
Homework Equations
3. The Attempt at a Solution [/B]
This is my reasoning, the function ##\operatorname...
Hi All,
$$\int{\exp((x_2x_1)^2+k_1x_1+k_2x_2)dx_1dx_2}$$
I can perform the integration of the integral above easily by changing the variable
$$u=x_2+x_1\\
v=x_2x_1$$
Of course first computing the Jacobian, and integrating over ##u## and ##v##
I am wondering how you perform the change of...
Homework Statement
dz/dx=(3x2+x)(2x^3+x^2)^2[/B]Homework Equations
∫(3x^2+x)(2x^3+x^2)^2 dx
The Attempt at a Solution
I tried substituting (2x^3+x^2)
Let t= 2x^3 + x^2
dt=6x^2+2x dx
dt/dx= 6x^2+2x
I can only solve till this point . I don't have any clue how to solve it further
But how do we...
Homework Statement
Find the integral \int \frac{3x+1}{(x^2x6)\sqrt{3x^2+4x7}}\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{1}{(x2)^3\sqrt{3x^28x+5}}\mathrm dx
2. The attempt at a solution
I can't find a useful substitution to solve this integral.
I tried x2=\frac{1}{u},x=\frac{1}{u}+2,dx=\frac{1}{u^2}du that gives
\int \frac{1}{(x2)^3\sqrt{3x^28x+5}}\mathrm...
Homework Statement
Let:
##I=\int _{1} ^{1}{\frac{dx}{\sqrt{1+x}+\sqrt{1x}+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...
Just for my own entertainment I integrated the Lorentz factor with respect to velocity, using basic trig sub, I got the equation arcsin(v/c)*(mc^2). What does this mean? Is it just useless and irrelevant in the physics world or does it have some sort of hidden meaning?
Hi,
I'm trying to solve ∫(x21)/(1+x2)*1/(1+x4)(1/2)dx
I'm apparently meant to get some noncomplex result, the question suggests to use the substitution u2 = x2 + 1/x2
But I haven't gotten anywhere with this.
Any methods or suggestions (or the solution) would be much appreciated!
Thanks
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...
Ok so I might be doing something silly but I just don't understand what is going on here. So the integral:
i = ∫ sin x (cos x)^3 dx
First I say u = cos x. So du =  sin x dx.
So now I have i = ∫  u^3 du. Which gives: i = (1/4)u^4 or (1/4)(cos x)^4. Easy.
But if I say u = sin x...
Homework Statement
∫1/(3+((2x)^.5))dx
the answer should be ((2x)^.5)  3ln(3+((2x)^.5)) + c
I keep getting ((2x)^.5)  ln(3+((2x)^.5)) + c
Homework Equations
∫1/(3+((2x)^.5))dx
The Attempt at a Solution
I did:
u = 3 + ((2x)^.5)
du = 1/((2x)^.5) dx
du((2x)^.5) = dx...
I am trying to compute the following integral:
\int \exp^{w^T \Lambda w}\, d\theta where \Lambda is a constant wrt \theta
w = y  t(x, \theta)
So, I am trying to use substitution and I have:
d\theta = \frac{dw}{t^{'}(x, \theta)}
So, substituting it, I have the following integral...
Homework Statement
I've been working on a problem from Apostol "Calculus" Volume 1 (not homework but self study). The problem is Section 5.8, Number 25 (Page 217) and states:
If [tex]$\m$[\tex] is a positive integer, show that:
\int_0^{\frac{\pi}{2}} cos^m x sin^m x dx =...
Homework Statement
Integrate the following using substitution techniques
∫e3tcsc(e3t)cot(e3t) dt
Homework Equations
csc(t) = 1/sin(t)
cot(t) = 1/tan(t)
cot(t) = cos(t)/sin(t)
1 + cot2(t) = csc2(t)
The Attempt at a Solution
∫e3tcsc(e3t)cot(e3t) dt
set u = cot(e3t)...
Homework Statement
Integrate 1/(1+x(sin(t))^2) between 0 and pi/2 using the substitution u = tan(t)The Attempt at a Solution
du/dt = (sec(t))^2
dt/du = 1/(1+u^2)
I've messed around with the integral and trig. identities but I don't seem to be getting anywhere changing the integral to make...
integration by parts
I'm working through Apostol's Calculus. I have attached the problem. I need to derive the formula integrating by parts.
It is not a hard problem, but I can't seem to understand how on Earth the author came up with that expression.
I take f(x) = (a^2  x^2)^n, so...
Homework Statement
From Larson, 9th Edition: Section 4.5. Solve the differential equation
\frac{\operatorname{d}y}{\operatorname{d}x}=4x+ \frac{4x}{\sqrt{16x^2}}Homework Equations
The Attempt at a Solution
Well, I can get my book's answer, but not through doing things in the prescribed way...
θHomework Statement
I'm trying to do an integration by substitution, but I'm completely stuck at the moment
∫(1sin2θ)cosθ dθ
Homework Equations
∫u dv = uv  ∫v du
The Attempt at a Solution
u = 1  sin2θ
dv = cosθ dθ
du = 2sinθcosθ or sin(2θ)
v = sin
I found du as...
Homework Statement
Calculate the following integral:
\int{\frac{\sqrt{x+1}}{x+5}dx} \ , x ≥ 1
By using the following substitution:
t=\sqrt{x+1}
Homework Equations
Well using the integration by substitution formula.
The Attempt at a Solution
So I have t=\sqrt{x+1}...
Homework Statement
Using a suitable substitution find the solution to:
∫ (x+2)50(x+1)dx
Homework Equations
The Attempt at a Solution
I can't find a solution to this using substitution. Wolfram alpha give an answer that is too long to be calculated by hand. Can anyone work...
Homework Statement
Original problem is differential equation dy/dx=(x+2y)/(3y2x)
This is part of solving differential equation.
x(dv/dx) = (1+4v3v^2)/(3v2)
so one way of solving, I take out the negative sign
x(dv/dx) = ((3v^24v1)/(3v2)) , separate and bring over...
The stupid question of the day.
Is it fair to say that\frac{du}{dx} = \frac 1 {dx/du}
since this comes (I think) from the chain rule,
\frac{dx}{du} \frac{du}{dx} = \frac{dx}{dx} = 1
Which means that, when integrating by substitution, I can choose to do either of
\int f(u) du = \int...
Homework Statement
To show that \int_{0}^ \frac{\pi}{2}\sqrt{cos\theta}sin^3(\theta) d\theta = 8/21
The Attempt at a Solution
The above expression was simplified as
\int_{0}^ \frac{\pi}{2}\sqrt{cos\theta}sin^2(\theta) sin(\theta) d\theta
\int_{0}^...
Homework Statement
a) Show that:
\int_{0}^{\pi} xf(sin (x))dx = \frac{\pi}{2}\int_{0}^{\pi} f(sin (x))dx
[Hint: u = π  x]
b) Use part a) to deduce the formula:
\int_{0}^{\pi} \frac{xsin(x)}{1 + cos^2 (x)} dx = \pi\int_{0}^{1} \frac{dx}{1 + x^2}
Homework Equations
\int_{a}^{b}...
Hi, I'm new to integration and I'm trying to figure out where I went wrong on this question. I'm close to the answer, but I can't tell where I've gone wrong? Can anyone help?
Thanks.
Homework Statement
Q. Evaluate the following:
The Attempt at a Solution
Please see attachment.
Homework Statement
I have the integral
\int sin(2.13\sqrt{x}+2.4)\,dx
I'm supposed to use the substitution y=2.13\sqrt{x}+2.4, aka. \sqrt{x}=\frac {y2.4}{2.13} to gain the following description of the integral:
\int sin(2.13\sqrt{x}+2.4)\,dx = E cos(y) + F\int y sin(y)\,dy
I have...
Homework Statement
Find the indefinite integral by substitution.
∫2x/(x+5)^6 dx
Homework Equations
The Attempt at a Solution
I know how to do this using the method of partial fractions, but the book says to use substitution. Is there a way to just do a basic usubstitution...
Hello all,
We've just begun integration in my maths class and I have a question about a certain aspect of integration by substitution.
Let's say for instance you let u = 2x1. Then you differentiate it and get du/dx = 2.
My maths teacher said " you can now think of it as multiplying...
Homework Statement
\int^2_1 6x\sqrt{x1}dxHomework Equations
The Attempt at a Solution
Let u=x1.
Then, u+1=x,
and du=dx.
Continued from problem statement,
=6 \int^1_0 (u+1)u^{\frac{1}{2}}du
=6 \int^1_0 u^{\frac{3}{2}} + u^{\frac{1}{2}}du
=6(1^{\frac{3}{2}} + 1^{\frac{1}{2}})
=6(2)
=12
My Web...
Hi , I solved \int (1/((\sqrt{(x^2)+(a^2)}))^(3/2)*dx) using the substitution x = a*tan(\varphi I wonder if there are other methods to solve this problem?
* (2/3) is the power on the radical function
Homework Statement
(i) find \int^{X}_{0} xe^{x^{2}} dx in terms of X.
(ii) Find \int^{X}_{0} xe^{x^{2}} dx for X= 1, 2, 3 and 4.
Homework Equations

The Attempt at a Solution
(i) \int^{X}_{0} xe^{x^{2}}dx
x^{2} = X
dX/dx=2x hence 1/2 dX = xdx
so, \frac{1}{2}...
Homework Statement
Integral of d.cos j with regard to d.sin j
Where d is a constant.
Homework Equations
The Attempt at a Solution
I don't know how to approach this. I can substitute u=d.sin j
Then I have
Integral of dz/dj with regard to dz, but not sure where to go from here.
Any help...
trying to solve the following integral by substitution but having trouble:
\int\cos^{5}7x\sin7xdx
I attempted to set u=\cos^{5}7x and ended up with (by chain rule...which I hope is correct!):
du=35\cos^{4}7x\sin7xdx
This doesn't seem too helpful but can't think of a better...
I am confused about integration in other cases, I understand that you can use substitution if the derivative exists next to what your trying to integrate then you can use it.
However while studying Arc Length and surface of a revolution I came across a problem such that I had to integrate the...
Homework Statement
indefinite integral 5\picos\pit
Homework Equations
The Attempt at a Solution
5\pi int cos\pit
Substitution Method
5\pi x sin (1/\pit
i don;t have a specific homework question. i have a sort of conceptual question instead
when integrating by substitution, how do i know what to choose as u?
for example
integral of z^2 / (1 + z^3)^(1/3) dz
i am suppposed to choose u as 1+z^3. any other value for u won't give me the...
Homework Statement
[PLAIN]http://img293.imageshack.us/img293/5026/solutoni.png
Hi all,
Can anyone explain what is going on where? I understand that it is a different way of writing the conventional integration by substitution, instead of using the symbol u. The second line, however...
Homework Statement
evaluate:
higher limit of 36
lower limit of 0 (36+3x)^1/2 dx
Homework Equations
i thought of using subsititution?
The Attempt at a Solution
g(x)=36+3x
g'(x)=3
when x=0, u=36+3(0)=36
when x=36, u=36+3(36)=144
from lower limit of 36 to higher...
Hi there,
I am having difficulty with one aspect of intergration by substitution where the substituion of a square root is U^2, wondering if anyone can help.
Problem:
Integral of: 2x√(3x4) dx by substituting U^2 = 3x4
Would du^2/dx = 3 therefore 1/3 du^2 = dx (I think...
Homework Statement
\int \frac{3x}{2x+3}
u = 2x +3
x = \frac{1}{2}(u3} )
dx = \frac{1}{2} du
so now the integral should be,
\int \frac{ \frac{3u9}{2}}{u} \times \frac{1}{2} du
= \frac{1}{2} \int \frac{3u9}{2} \times \frac{1}{u} du
\frac{1}{2} \int...
I'm attempting to solve the following problem:
\int_{0}^{\infty} {\frac{x arctan(x)}{(1+x^{2})^{2}}dx}
I started with a substitution:
u=arctan(x), du=\frac{1}{(1+x^{2})}dx
This seemed like the right thing to do, but after trying to put it together in several different ways I got...
Homework Statement
If f is continuous and \int^{9}_{0}f(x)dx = 4, find \int^{3}_{0}xf(x^{2})dx
Homework Equations
None required
The Attempt at a Solution
Don't really know where to begin, but I tried:
for \int^{3}_{0}xf(x^{2})dx
let:
u = x^{2}
du = 2xdx
substitute...
Hi, am I on the right track with this Usubstitution problem?
Homework Statement
Evaluate the indefinite integral
Homework Equations
integral of x^2(x^3 + 5)^9 dx
The Attempt at a Solution
integral of x^2(x^3 + 5)^9 dx
Let u = x^3 + 5
du = 2x^2
1/2du = x^2
1/2 integral u^9 du
1/2...