What is Integration by parts: Definition and 437 Discussions
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation.
The integration by parts formula states:
{\displaystyle dv=v'(x)dx}
, the formula can be written more compactly:
∫
u
d
v
=
u
v
−
∫
v
d
u
.
{\displaystyle \int u\,dv\ =\ uv\int v\,du.}
Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715. More general formulations of integration by parts exist for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called summation by parts.
In calculating the integral ##\int{\ln\left(x\right)\,\sin\left(x\right)\,\cos\left(2\,x\right)}{\;\mathrm{d}x}##, I used a few online integral calculators to check my answer. According to one calculator, I got the correct antiderivative, but according to another (Math DF Integral Calculator)...
I was just going through my old notes on this i.e
The concept is straight forward only challenge phew :cool: is the integration bit...took me round and round a little bit... that is for ##A_n## part.
My working pretty ok i.e we shall realize the text solution. Kindly find my own working...
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##...
Hi, PF
Integration by parts is pointed out this way:
Suppose that ##U(x)## and ##V(x)## are two differentiable functions. According to the Product Rule,
$$\displaystyle\frac{d}{dx}\big(U(x)V(x)\big)=U(x)\displaystyle\frac{dV}{dx}+V(x)\displaystyle\frac{dU}{dx}$$
Integrating both sides of...
Using integration by parts:
$$I_n=\left. x(1+x^2)^{n} \right_0^1+\int_0^{1} 2nx^2(1+x^2)^{(n+1)}dx$$
$$I_n=2^{n} + 2n \int_0^{1} x^2(1+x^2)^{(n+1)}dx$$
Then how to continue?
Thanks
I'm trying to compute ##\int_0^1 x^m \ln x \, \mathrm{d}x##. I'm wondering if the bit about the application of L'Hopital's rule was ok. Can anyone check?
Letting ##u = \ln x## and ##\mathrm{d}v = x^m##, we have ##\mathrm{d}u = \frac{1}{x}\mathrm{d}x ## and ##v = \frac{x^{m+1}}{m+1}##...
How can i move from this expression:
$$\frac{4}{\pi^{4}} \int dk \frac{1}{k^2} \frac{1}{(1+i(kk_{f}))^3} \frac{1}{(1+i(kk_{i}))^3}$$
to this one:
$$\frac{4}{\pi^{4}} \int dk \frac{1}{k^2} \frac{1}{(1+kk_{i}^2)^2} \frac{1}{(1+kk_{f}^2)^2}$$
using Feynman parametrization (Integration by...
I simply plugged \phi = \phi_0 (\eta) + \delta \phi (\eta, \vec x) into the given action to get
\begin{align}
S &= \int d^4 x \left[ \frac{a^2}{2}\left(\dot \phi^2 (\nabla \phi)^2\right)a^4V(\phi) \right] \nonumber \\
&= \int d^4 x \left[ \frac{a^2}{2}\left(\dot \phi_0^2 + (\delta...
As we all know, integration by parts can be defined as follows: $$\int u dv = uv  \int v du$$ And the usual strategy for solving problems of these types is to intelligently define ##u## and ##dv## such that the RHS integral can easily be evaluated. However, something that is never addressed is...
So I am confused about a proof in which the formula for expected value of velocity, ##\frac{d\langle x \rangle}{dt} ##, is derived.
Firstly, because the expected value of the position of wave function is $$\langle x \rangle =\int_{\infty}^{+\infty} x\Psi(x,t)^2 dx$$Therefore...
I'm reading Coleman's "Aspects of symmetry" chap 7.
On the topic of the SU(2) winding number on ##S^3##on page 288, three parameters on ##S^3## are defined ##\theta_1,\theta_2,\theta_3##. Afterwards, it defines the winding number and to show it's invariant under continuous deformation of gauge...
I think in the case of "n da" you can see the denominator (1+x^2) as a constant, so
∫ ( sin(a) + M^2 ) / ( 1 + x^2 ) da
= ( 1 / ( 1 + x^2 ) ) * ∫ (sin(a) + M^2 ) da
= ( 1 / ( 1 + x^2 ) ) * ( cos(a) + (M^2)a )
= (  cos(a) + (M^2)a ) / ( 1 + x^2 )

Is this the way to go? This is my...
Part of me thinks this is could be a usub b/c x^3's derivative is 3x^2, a factor of 3 off from what e is raised to...but it is not a traditional usub...any thoughts if this is a usub or by parts, and what u should be?I know that there is more to solving the equation after this ( z =...
Hello, I'm just starting Zee's QFT in a Nutshell, I'm a bit confused about what he means by "integate by parts under the d4x". Can someone explain what he means by this? I understand how to obtain the KleinGordon equation from the free particle Lagrangian density, but not sure why he invokes...
The book on quantum mechanics that I was reading says:
d<x>/dt = d/dt ∫∞∞ ψ(x,t)2 dx
=iħ/2m ∫∞∞ x∂/∂x [ψ∂ψ*/∂x+ψ*∂ψ/∂x]dx (1)
=∫∞∞ [ψ∂ψ*/∂x+ψ*∂ψ/∂x]dx (2)
I want to know how to get from (1) to (2)
The book says you use integration by part:
∫abfdg/dx dx = [fg]ab  ∫abdf/df dg dx
I chose f...
Homework Statement
I want to integrate
∫e^(2x)*sin(e^x) dx
Homework Equations
∫uv'dx=uv  ∫u'v
The Attempt at a Solution
u = e^2x
du = 2*e^2x
dv = sin(e^x)
v = cos(e^x)/e^x
e^(x)*cos(e^x)  2∫e^(x)*cos(e^x) dx
e^(x)*cos(e^x)  2*sin(e^x) + c
The solution I have doesn't have the two in...
Homework Statement
∫e^(x)(1tanx)secx dx
2. Attempt at a solution
I know ∫e^x(f(x)+f'(x))=e^x f(x)
and I intuitively know f(x) could be secx here and therefore f'(x) will be secxtanx but I can't figure out how to reach that

check if right
check if right
Now, 2 seems to be the right answer for A yet when i made x=5 and subtracted new form form the old one I got a difference of ~$\frac{4}{9}$ (should be 0 obviously) I got A=2 B=$\frac{45}{21}$ C=2
How to calculate $\lim_{{x}\to{\infty}}( e^{x})$
Hi,
I've been following a derivation of relativistic kinetic energy. I've seen other ways to get the end result but I'm interested in finding out where I've gone wrong here: I'm struggling with integrating by parts.
The author goes from...
Hello all,
I am trying to understand the rational behind the visualization of integration by parts, however I struggle with it a wee bit.
I was trying to read about it in Wiki, this is what I found...
I'll cut the long story short. What on Earth happened here:
I seem to be unable to do the integration by parts of the first term. I end up with a lot of dx's.
Homework Statement
##\displaystyle \int \frac{\log (x)}{x}~ dx##
Homework EquationsThe Attempt at a Solution
I am a little confused about the first part. We know that the ##\displaystyle \int \frac{1}{x}~ dx = \log x##. So how can we proceed with integration by parts if one of the logs has...
Homework Statement
Integrate e^3x sin x.
Homework Equations
uv  Integral(v du)
The Attempt at a Solution
I am trying to help somebody else with this problem, as I took Calculus a few years ago, but the end is really kicking my butt. I know I'm VERY close, but once I get to the second...
I am reviewing Jackson's "Classical Electromagnetism" and it seems that I need to review vector calculus too. In section 1.11 the equation ##W=\frac{\epsilon_0}{2}\int \Phi\mathbf \nabla^2\Phi d^3x## through an integration by parts leads to equation 1.54 ##W=\frac{\epsilon_0}{2}\int \mathbf...
Homework Statement
I am trying to show that ## \int \delta (xa) \delta (xc) dx = \delta (ac) ## via integeration by parts, but instead I am getting ##\delta (ca) ## (or ##\delta (ac)## depending how I go...).
Can someone please help me out where I've gone wrong: struggling to spot it...
Question: sqrt(x) cos(sqrt(x)) dx
My try:
Let dv = cos(√x) => v = 2√xsin(√x) and u = √x => du = dx/(2√x)
Using integration by parts, we get
∫√x cos(√x) dx = 2√x√x sin(√x)  ∫(2√xsin(√x) dx)/(2√x)
= 2x sin(√x)  ∫sin(√x) dx
= 2x sin(√x) + 2 cos(√x) √x
However, the answer given in the book...
Hello, I'm currently taking calc 1 as an undergraduate student, and my professor just showed us a new? way of solving Integration By Parts.
This is the example he gave"
Is there a name for this technique that substitutes d(___) instead of dx?
Thank you,
Homework Statement
Hi, I'm doing a variation of parameters problem for my differential equations class. It requires solving the integral:
∫ex t2 dt
I am sure my professor did not give me an impossible integral and that there is some algebraic "trick" to solving it, but despite going through...
Homework Statement
An object of mass m is initially at rest and is subject to a timedependent force given by F = kte^(λt), where k and λ are constants.
a) Find v(t) and x(t).
b) Show for small t that v = 1/2 *k/m t^2 and x = 1/6 *k/m t^3.
c) Find the object’s terminal velocity.
Homework...
Hello  I'm not sure this is where this should go, but I'm working with Laplace Transforms and differential equations, so this seems as good a place as any. Also, I doubt this is graduate level math strictly speaking, but I went about as high as you can go in calculus and linear algebra during...
Homework Statement
[/B]
Homework Equations
∫ f(x) g'(x) dx = f(x) g(x)  ∫ f '(x) g(x) dx
f(x)=√(1+x^2)
f '(x)=x * 1/√(1+x^2)
g'(x)=1
g(x)=x
The Attempt at a Solution
∫ √(1+x^2) * 1 dx
=x * √(1+x^2)  ∫ x^2 * 1/√(1+x^2) dx
Further integration just makes the result look further from what...
I was trying to derive the following results from 4B.8 as suggested by using the vector triple product identity but have been unsuccessful in deriving ##\vec{L_R}## and ##\vec{S_R}## in the end. After using the identity and finding the integrand to be ## \vec{E}(\vec{r}\cdot\vec{B})  \vec{B}...
I'am trying to prove
\int e^{ix}cos(x) dx= \frac{1}{2}x\frac{1}{4}ie^{2ix}
Wolfram tells so http://integrals.wolfram.com/index.jsp?expr=e^%28i*x%29cos%28x%29&random=false
But I am stuck in obtaining the first term:
My step typically involved integration by parts:
let u=e^{ix}cos(x) and...
This was just very basic, I have accepted it in just a heartbeat, but when I tried to chopped it and examined one by one, somethings fishy is happening, this just involved \int_{0}^{\infty}x e^{x}dx=1.
Well, when we do Integration by parts we will have let u = x du = dx dv = e^{x}dx v =...
Hi all,
I have the average value of a function between limits of 7.3826 and 0 which equals 0.4453. I have used the formula for average value function and attached the equation I need solving as I don't know how to use the Latex commands. P is what I am trying to work out. Unfortunately I have...
Homework Statement
Integrate $$\int_0^1 dw \frac{w^{\epsilon+1} \ln((r+1w)/r)}{1+r(1+w)}$$ for ##\epsilon## not necessarily an integer but positive and r is negative (<1). The argument of the log is positive.
Homework Equations
Integration by parts
The Attempt at a Solution
[/B]
I can...
In this thread, ramparts asked how integration by parts could be used in general relativity.
suppose you have
##\int_M (\nabla^a \nabla_a f) g .Vol##
Can it be written like
##\int_M (\nabla^a \nabla_a g) f .Vol## plus a boundary integration term (by integrating twice by parts)?
I think thay it...
In this paper
we have p18 an integral on space time M. The author takes a 3 dimensional space like Cauchy surface ##\Sigma## which separates M in two regions, the future and the past of ##\Sigma##. He gets so the sum of two integrals on these regions. He writes then let us integrate each of them...
The Integral:
\int{\sin{(\theta)}*\cos{(\theta)}*d\theta}
Attempt to solve by Integration by Parts:
\int{u*dv} = u*v  \int{v*du}
u = \sin{(\theta)}
du = \cos{(\theta)}*d\theta
v = \sin{(\theta)}
dv = \cos{(\theta)}*d\theta
Bringing back to the beginning.
Using the MaxwellBoltzmann equation above, there is an example in my book (Giancoli 4th edition p. 481) where they use this to find the average velocity. I understand that it would just be the sum of all the speeds of the molecules divided by the number of molecules. But then I'm having...
Find the second degree polynomial P(x) that has the following properties: (a) P(0)=1, (b) P'(0)=0, (c) the indefinite integral ∫P(x)dx/(x^3(x1)^2). Note: the the indefinite integral is a rational function. Cannot have Log terms occurring in solution.
first. I use the generic polynomial...
Homework Statement
While integrating by parts( by the formula) why don't we consider the contant of integration for every integral in the equation.
Homework Equations
$$∫uv = u∫v  ∫ ∫v . d/dx(u) $$
The Attempt at a Solution
[/B]
example.
$$∫x \sin(x) dx = ?? $$
this is can be done like...
I have the following integral \int e(2x) cos(ex).
Let u = ex
Do integration by parts:
\int u2cos(u) du = u2sin(u)  \int (2usin(u) du
Do integration by parts again for \int (2usin(u) du:
\int (2usin(u) du = 2ucos(u)  \int 2cos(u) du
Putting it all together:
\int e(2x) cos(ex) =...
Homework Statement
I'm going to cut from the initial part of the problem, which I am confident is good to go, and cut straight to the antiderivatives.
Homework Equations
All antiderivatives are to be integrated on the interval from 0 to π/18
(I1) = 1/9 cos 9x  (I2) (2/27 * cos3(9x)) + (I3)...