Ilm Gah Website is under maintenance! If you found any error please let us know. Contact Us ×

Integration by Parts is a special method of integration that is often useful when two functions are multiplied together, but is also ...

Integration by Parts is a special method of integration that is often useful when two functions are multiplied together, but is also helpful in other ways.
You will see plenty of examples soon, but first let us see the rule:
u v dx = uv dx −u' (v dx) dx
• u is the function u(x)
• v is the function v(x)
As a diagram:
Let's get straight into an example, and talk about it after:

Example: What is ∫x cos(x) dx ?

OK, we have x multiplied by cos(x), so integration by parts is a good choice.
First choose which functions for u and v:
• u = x
• v = cos(x)
So now it is in the format u v dx we can proceed:
Differentiate uu' = x' = 1
Integrate vv dx = cos(x) dx = sin(x)   (see Integration Rules)
Now we can put it together:
Simplify and solve:
x sin(x) − sin(x) dx
x sin(x) + cos(x) + C

So we followed these steps:
• Choose u and v
• Differentiate u: u'
• Integrate v: v dx
• Put u, u' and ∫v dx into: uv dx −u' (v dx) dx
• Simplify and solve
(u integral v) minus integral of (derivative u, integral v)

Let's try some more examples:

Example: What is ∫ln(x)/x2 dx ?

First choose u and v:
• u = ln(x)
• v = 1/x2
Differentiate u: ln(x)' = 1/x
Integrate v: 1/x2 dx = x-2 dx = −x-1 = -1/x   (by the power rule)
Now put it together:
Simplify:
−ln(x)/x − ∫1/x2 dx = −ln(x)/x − 1/x + C
−(ln(x) + 1)/x + C

Example: What is ∫ln(x) dx ?

But there is only one function! How do we choose u and v ?
Hey! We can just choose v as being "1":
• u = ln(x)
• v = 1
Differentiate u: ln(x)' = 1/x
Integrate v: 1 dx = x
Now put it together:
Simplify:
x ln(x) − 1 dx = x ln(x) − x + C

Example: What is ∫ex x dx ?

Choose u and v:
• u = ex
• v = x
Differentiate u: (ex)' = ex
Integrate v: x dx = x2/2
Now put it together:
Well, that was a spectacular disaster! It just got more complicated.
Maybe we could choose a different u and v?

Example: ∫ex x dx (continued)

Choose u and v differently:
• u = x
• v = ex
Differentiate u: (x)' = 1
Integrate v: ex dx = ex
Now put it together:
Simplify:
ex − ex + C
ex(x−1) + C
The moral of the story: Choose u and v carefully!
Choose a u that gets simpler when you differentiate it and a v that doesn't get any more complicated when you integrate it.
A helpful rule of thumb is I LATE. Choose u based on which of these comes first:
• I: Inverse trigonometric functions such as sin-1(x), cos-1(x), tan-1(x)
• L: Logarithmic functions such as ln(x), log(x)
• A: Algebraic functions such as x2, x3
• T: Trigonometric functions such as sin(x), cos(x), tan (x)
• E: Exponential functions such as ex, 3x

And here is one last (and tricky) example:

Example: ∫ex sin(x) dx

Choose u and v:
• u = sin(x)
• v = ex
Differentiate u: sin(x)' = cos(x)
Integrate v: ex dx = ex
Now put it together:
ex sin(x) dx = sin(x) ex -cos(x) ex dx

Looks worse, but let us persist! We can use integration by parts again:
Choose u and v:
• u = cos(x)
• v = ex
Differentiate u: cos(x)' = -sin(x)
Integrate v: ex dx = ex
Now put it together:
ex sin(x) dx = sin(x) ex - (cos(x) ex −−sin(x) ex dx)
Simplify:
ex sin(x) dx = ex sin(x) - ex cos(x) − ex sin(x)dx
Now we have the same integral on both sides (except one is subtracted) ...
... so bring the right hand one over to the left and we get:
2ex sin(x) dx = ex sin(x) − ex cos(x)
Simplify:
ex sin(x) dx = ex (sin(x) - cos(x)) / 2 + C

Footnote: Where Did "Integration by Parts" Come From?

It is based on the Product Rule for Derivatives:
(uv)' = uv' + u'v
Integrate both sides and rearrange:
(uv)' dx = uv' dx + u'v dx
uv = uv' dx + u'v dx
uv' dx = uv − u'v dx
Some people prefer that last form, but I like to integrate v' so the left side is simple:
uv dx = uv dx − u'(v dx) dx

Name

Admissions,7,Agri-Jobs,1,Animated Moral Stories,2,Atomic-Jobs,1,California Scholarships,1,Canada Scholarships,2,Colleges,1,Computer,32,Conferences,5,CSS Agriculture,3,CSS Current Affairs,4,CSS Data,62,CSS Date Sheet,1,CSS English Essay,10,CSS Everyday Science,7,CSS General Knowledge,1,CSS History of Pakistan,2,CSS Islamiyat,4,CSS Pakistan Affairs,28,CSS Past Paper,59,Czech Scholarships,1,Data,5,Date Sheets,5,Derivative,2,Differentiate between,2,Differentiation,1,Employees,1,English1 FSc Notes,15,English3 FSc Notes,3,English4 FSc Notes,2,Entry Test,1,Essay Writing,3,Exchange Programs,3,Fellowships,4,firdous,1,Food Jobs,2,Form of Verbs,26,Fully Funded Scholarships,16,Funded,1,Germany Scholarships,1,Guess 2nd Year,10,HEC HAT Test,1,Important Questions Chemistry2 FSc,4,Important Questions Physics2 FSc,5,Integration,5,Inter Date Sheet,1,Internships,6,Italy Scholarships,1,Japan Scholarships,2,Jobs,15,Masters Scholarships,16,Moral-Stories,1,NAB-Jobs,1,Netherland Scholarships,1,Netherlands Scholarships,1,News-Updates,6,Novel,1,OP Questions Chemistry2 FSc,4,OP Questions Physics2 FSc,5,Partially Funded,2,PhD Scholarships,8,Presentations,2,Quotes,1,Results,12,Roll Number Slips,1,Scholarships,30,Singapore Scholarships,1,South Korea Scholarships,2,Statistics,6,Switzerland,1,Test Chemistry2 FSc,1,Tips,3,Training Programs,1,Turkey Scholarships,1,Undergraduate Scholarships,7,United Kingdom,2,Urdu Essays,1,USA Scholarships,4,Vacancies,1,Videos,2,Writting Tips,1,
ltr
item
Ilm Gah: Integration by Parts
Integration by Parts