# Calculus demystified for Real People, Part 4 -- Formulae Rearranged

**Integration By Parts**

Since we started to study ourselves this crazy math..we wanted to find ways to simplify 'Heuristically' using easier Methods and our own Mnemonics instilled in us by our Tutor... That our Dad got us in Junior High. Lucky us right?

Using some rules of Calculus and rearranging formulas we can attack more complicated problems that will arise in the life of a College dude, you know?

**STRAIGHT TO our Subject**

The Equation to the right has a derivative g′ that could be a handicap since we usually see more down to earth functions that multiply themselves as in f(x).g(x)

If we really need a practical formula for

## ∫f(x).g(x)dx

We postulate a new arrangement as in the formula below. You can obtain this new formula by Integrating both sides of the equation

We've got to make sure that f can be differentiated to f′

and **g** be able to get integrated as well...

**EXAMPLE :**

**We should know that ∫(1/x**^{2})dx = -1/x

^{2})dx = -1/x

**and ∫x**^{-1}dx = ln/x/ +c

^{-1}dx = ln/x/ +c

**Consequently Inversely , we can asses that:**

## (ln x) ′= 1/x ( differentiation of natural logarithm for x)** **

Let's say we want to integrate this function....we know beforehand that we can break it down...

We rearrange it to our formula of 2 functions

∫(lnx)(x^{-2 })dx= lnx ∫x^{-2}dx- ∫ f′ (∫g)dx

where f′ equals to 1/x

and g integrated (∫g) equals to -1/x

replacing in the formula for integration of two functions of the type ∫f.gdx=??

we get the easier to solve equation:

**The answer is**

**-lnx/x -1/x**

**or -(lnx+1)/x**

**You should know by now that**

**∫x^k or ∫x ^{k }= x^{k+1}/k+1**

**and inversely**

** (x ^{k})′= kx^{k-1 }(differentiation of a power)**

****

****

## Comments

Well.. we now know who the genius of the group is. :) I was going to check useful but who am I kidding. I will pass this on to my son who is an engineering major and in calculus now. Well done, beautiful fonts!

Well, my Lord, I gave you an awesome on this one. It must be awesome. It's Greek to me, though. I'm amazed at your talents--Calculus, poetry, and you even threw in a few recipes of late....I'm sure this hub will be very helpful to the masses who study calculus. As for me, I think I'll go over to one of your algebraic hubs, instead. :-) Good job! I am duly impressed.

Hey Lord,

Some nice ratings and the hub will be good for a while as Calculus hasn't changed since Euclid. I have my Calculus book for Engineers in the car. I don't remember this part of Calculus but it has been a few years. I'll have to go back and look it over.

Great hub. Very educational and it will inspire a lot of young people to take higher advanced Math classes!!

Awesome and Vote Up!! JT

Lord,

You are doing a great service to the kids!! Man you are prolific.

JT

6