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Questions tagged [integration-by-parts]

To be used when the technique of Integration By Parts is the dominant topic of the question.

The integration by parts technique is used frequently. The method is used when integrating the product of functions by using an identity that is the result taking the integral of the multiplication rule for derivatives. The main goal of it is to change the integration so that one of the functions inside is integrated while the other is differentiated. Repeated application is intended to make one of the functions reduce to a constant, while having the other function be something that loops as it is repeatedly integrated, such as $\sin(x)$, $\cos(x)$, and $e^x$

The term LIATE is generally used to determine which item should be differentiated. Higher items should take precedence:

  • L - logarithmic functions
  • I - inverse trigonometric functions
  • A - algebraic functions
  • T - trigonometric functions
  • E - exponential functions

An outline of the proof for integration by parts is given as follows:

Take the multiplication rule for derivatives:

$$\frac {\mathrm{d}}{\mathrm{d}x}f(x)g(x) = f'(x)g(x) + f(x)g'(x)$$

Shift terms around:

$$f'(x)g(x) = \frac {\mathrm{d}}{\mathrm{d}x}f(x)g(x) - f(x)g'(x)$$

Integrate both sides:

$$\int f'(x)g(x) dx = f(x)g(x) - \int f(x)g'(x) dx$$

And that final line is the identity known as integration by parts.

229 questions
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Integration by parts: the variance of a standard normal

I am calculating the variance of a standard normal, but I stuck with the following part (the answer is different from what I know). What is wrong with my calculation? $$ \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} y^2 e^{- y^2 / 2} = \left[ y^2…
user51966
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Integration by parts of the $\Gamma(x+1)$ doesn't match answer in book Pattern Recognition and Machine Learning

I'm working through problem 1.17 in Pattern Recognition and Machine Learning where I'm getting: \begin{align*} \Gamma(x+1) = & \int_{0}^{\infty} u^{(x+1)-1}e^{-u} \hspace{1mm} du \\ = & \big[-u^x e^{-u} \big]_{0}^{\infty} - \int_{0}^{\infty}…
Matt
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Integration by parts with unity factor

Famously, log and arctan can be integrated by parts when you write $\log(x)=\log(x)\cdot1$. Are there any other somewhat elementary examples of that phenomenon?
user413365
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Recursion in Integration by Parts

I'm trying to integrate by parts, but I keep getting recursive answers. $I=\int_0^\pi f(x)cos(x)dx$ where $f''(x)=3f(x)$, $f'(0)=-5$, and $f'(\pi)=4$ Thanks.
sparky
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I would like a hint for solving this indefinite integral

According to the Wolfram integrator: $$\int (x^2+a)^{-3/2}\text{d}x = {x \over a \times \sqrt{(x^2+a)}}$$ I easily differentiated the answer to verify that it was correct (not that I don't trust Wolfram or anything), but how would I get this result…
pmennen
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How to deal with derivatives with integration by parts

I am dealing with the derivative which is part of a basic weak formula (with the weighted $v$ and solution $u$): $$\int_{0}^{1}\left(-v(x+2)\frac{d^2u}{dx^2}+vu\right)\,dx$$ $$\Big(\frac{du}{dx}\Big)\Big|_{x=0}=3$$$$ u(1)=2$$ I know that I will…
Asinine
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Integrating $\int_a^b dx e^{-c\sinh ^2 x}$ by parts

I'm having trouble finding a reasonable way to integrate the following by parts: $$\int_a^b dx \ e^{-c\sinh ^2 x}$$ The idea is to get an expansion in powers of $c$ (I have to later show that this expansion doesn't work in fact), but I have no clue…
Spine Feast
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Integration by parts for $\tanh(x)$

Can you integrate $\tanh(x)$ by parts? I realize it's probably easier to use a derivative substitution, but can it be done? Essentially I just want some clarification that I'm using integration by parts correctly. $u=\tanh(x)$, $u'=sech^2(x)$ $v=x$,…
lost
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