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:

a

b

u
(
x
)

v

(
x
)

d
x

=

[

u
(
x
)
v
(
x
)

]

a

b

a

b

u

(
x
)
v
(
x
)

d
x

=
u
(
b
)
v
(
b
)

u
(
a
)
v
(
a
)

a

b

u

(
x
)
v
(
x
)

d
x
.

{\displaystyle {\begin{aligned}\int _{a}^{b}u(x)v'(x)\,dx&={\Big [}u(x)v(x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)v(x)\,dx\\[6pt]&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx.\end{aligned}}}
Or, letting

u
=
u
(
x
)

{\displaystyle u=u(x)}
and

d
u
=

u

(
x
)

d
x

{\displaystyle du=u'(x)\,dx}
while

v
=
v
(
x
)

{\displaystyle v=v(x)}
and

d
v
=

v

(
x
)
d
x

{\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.

View More On Wikipedia.org
1. I Did Math DF's integral calculator make a glaring mistake?

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)...
2. Finding the Fourier cosine series for ##f(x)=x^2##

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...
3. B Integration by parts of inverse sine, a solved exercise, some doubts...

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{1-x^2},\quad{V=x}## ##=x\sin^{-1}x-\int \frac{x}{\sqrt{1-x^2} \, dx}## Let ##u=1-x^2##...
4. B Integration by Parts, an introduction I get confused with

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...
5. Solving this definite integral using integration by parts

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
6. I Verifying Integration of ##\int_0^1 x^m \ln x \, \mathrm{d}x##

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}##...
7. A Feynman parametrization integration by parts

How can i move from this expression: $$\frac{4}{\pi^{4}} \int dk \frac{1}{k^2} \frac{1}{(1+i(k-k_{f}))^3} \frac{1}{(1+i(k-k_{i}))^3}$$ to this one: $$\frac{4}{\pi^{4}} \int dk \frac{1}{k^2} \frac{1}{(1+|k-k_{i}|^2)^2} \frac{1}{(1+|k-k_{f}|^2)^2}$$ using Feynman parametrization (Integration by...
8. Rewriting a given action via integration by parts

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...
9. B Why don't we account for the constant in integration by parts?

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...
10. I How to interpret integration by parts

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...
11. Integration by parts on ##S^3## in Coleman's textbook

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...
12. Integrating with a Denominator of (1+x^2): A Step-by-Step Guide

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...
13. Integration problem using Integration by Parts

i would appreciate alternative method...
14. I Integrate 1/(x*lnx): Integration by Parts

can integrate 1/(x*lnx) by parts??
15. I Bernoulli Equation with weird integral

Part of me thinks this is could be a u-sub 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 u-sub...any thoughts if this is a u-sub or by parts, and what u should be?I know that there is more to solving the equation after this ( z =...
16. A Understanding Integration by Parts in Quantum Field Theory

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 Klein-Gordon equation from the free particle Lagrangian density, but not sure why he invokes...
17. I Solving Quantum Mechanics Integral Equation: How to Get from (1) to (2)?

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...
18. Integration by parts -- help please

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...
19. What is the primitive of sinx/cos^2x?

Homework Statement ∫e^(-x)(1-tanx)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
20. MHB Integration by parts, Partial fraction expansion, Improper Integrals

- 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 $\lim_{{x}\to{\infty}}(- e^{-x})$
21. I Solving Integration by Parts for Relativistic Kinetic Energy

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...
22. MHB Visualization of Integration by Parts

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...
23. A Integration by parts of a differential

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.
24. Integration by Parts with Logarithmic Functions

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...
25. MHB Integration By Parts: uv-Substitution - 9.2

$\tiny{9.2}$ \begin{align*} \displaystyle I&=\int y^3e^{-9y} \, dx\\ \textit{uv substitution}\\ u&=y^3\therefore \frac{1}{3}du=y^2dx\\ dv&=e^{-9y}\, dx\therefore v=e^{-9y}\\ \end{align*} will stop there this looks like tabular method better
26. Integration by Parts Twice: How to Solve Tricky Integrals

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...
27. I Laplacian in integration by parts in Jackson

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...
28. Delta property, integration by parts, heaviside simple property proof

Homework Statement I am trying to show that ## \int \delta (x-a) \delta (x-c) dx = \delta (-a-c) ## via integeration by parts, but instead I am getting ##\delta (c-a) ## (or ##\delta (a-c)## depending how I go...). Can someone please help me out where I've gone wrong: struggling to spot it...
29. I Integrating sqrt(x) cos(sqrt(x)) dx

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...
30. I Integration by Parts without using u, v

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,
31. Solve Difficult Integral: ∫ex t-2 dt

Homework Statement Hi, I'm doing a variation of parameters problem for my differential equations class. It requires solving the integral: ∫ex t-2 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...
32. Integration by parts and approximation by power series

Homework Statement An object of mass m is initially at rest and is subject to a time-dependent 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...
33. A Triple Product in Laplace Transform

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...
34. MHB Why does the integral of √(a² +x²) need Integration by parts?

Why this integral $\int\left\{\sqrt{{a}^{2}+{x}^{2}}\right\}dx$ needs integration by parts? Thanks Cbarker1

49. MHB Integration by Parts: $\int u\cos(u)\,\mathrm{d}u$

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) =...
50. Integration by parts, just need a small hand

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)...