Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

Business Mathematics and Statistics

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B Com : Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

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Integration by Parts

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:
Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

  • u is the function u(x)
  • v is the function v(x)

As a diagram
Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev
 Let's get straight into an example, and talk about it after:

Example: What is Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev ?

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 Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRevwe can proceed:

Differentiate u: u' = x' = 1

Integrate vIntegration by parts method, Business Mathematics & Statistics B Com Notes | EduRev   (see Integration Rules)

Now we can put it togethe

Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

 

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:Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev
  • Put u, u' and ∫v dx into:u Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev
  • Simplify and solve

In English, to help you remember, Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRevbecomes:

(u integral v) minus integral of (derivative u, integral v)

Let's try some more examples:

Example: What is  Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

First choose u and v:

  • u = ln(x)
  • v = 1/x2

Differentiate u: ln(x)' = 1/x

Integrate v: Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev  (by the power rule)

Now put it together:

Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

Simplify:

− ln(x)/x − Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

− (ln(x) + 1)/x + C

 Example: What is Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

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:Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

Now put it together:

Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

 

Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

 Example: What is   Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

Choose u and v:

  • u = ex
  • v = x

Differentiate u: (ex)' = ex

Integrate Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

Now put it together: 

Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

Well, that was a spectacular disaster! It just got more complicated.

Maybe we could choose a different u and v?
 Example:   Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

Choose u and v differently:

  • u = x
  • v = ex

Differentiate u: (x)' = 1
 Integrate   Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

Now put it together:

Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

Simplify:

x 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: Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

  • u = sin(x)
  • v = ex

Differentiate u: sin(x)' = cos(x)
 Integrate   Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

Now put it together:
Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev    

Simplify:

Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

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:
Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev
 Simplify:
Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

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:

Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev 

Some people prefer that last form, but I like to integrate v' so the left side is simple:

Integration by parts method, Business Mathematics & Statistics B Com Notes | EduRev

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