Warning! Never try to read this article on calculations
Do you think that the calculations are Latin? So here is the output to calculate your strength!
This topic covered a brief discussion on Integration, an important branch of calculus. From the historical point of view, integration means the addition of infinite series whose terms are infinitesimally small. Integral calculus is the branch of mathematics we use to find lengths, areas, and volumes of irregular shapes.
We will now define integration mathematically. Let f ( x ) be a given
function of x. If we can determine another function F ( x ) , such that its derivative with respect to x is equal to f ( x ) [ differential of F ( x ) is equal to f ( x ).dx . Then F ( x ) is
defined as an indefinite integral of f ( x ) .
It is not possible to shed light on the whole part of integration by a single article . So ,today let’s start discussing on Integration by Parts method .It is the method of integration when we are to integrate product of two functions .
If u and v are two differentiable function of x , then ,
∫ uv d x = u ∫ dv – [ [ du / dx ∫ v dx ] dx
that is to say . the integral of the product of two functions = [ ( first function × integral of second function ) – ( integral of ( derivative of 1st function × integral of second function ]
But we have to remember the rule to select the function choice order. The meaning of the word ILATE is described below:
I –denotes inverse function
L – denotes logarithmic function
A -denotes algebraic function
T- denotes trigonometric function
E -denotes exponential function
Now let’s try to solve the following simple problems:
( 1 ) ∫ xe^ x dx = x ∫ e^ x dx – ∫ [d/dx ( x ) ∫ e^x dx ] dx
|Take u = x ( algebraic function ) |
| yv = e^x ( exponential function ) |
= xe^x – ∫ e ^ x dx | d/dx(x)=1 |
= xe ^ x – e ^ x + C | C is an arbitrary constant
= ( x – 1 ) e ^x + C
( 2 ) Find ∫ e ^ x Sin x dx
Let I = ∫Sin xe ^ x dx | Take u = Sin x (trigonometric function)
| yv = e^x ( exponential function )
= Sin x ∫ e ^ x dx – ∫[ d /dx ( Sin x ) ∫ e ^ x ] dx
= Sin xe ^x – ∫ [Cos x e ^ x ] dx
| Again we have to apply By Parts to solve ∫Cosx e^xdx
= sin xe ^ x – [ Cos x ∫ e ^ x – ∫ [d / dx ( Cos x ) ∫ e^ x ] dx
= sin xe ^ x -[ Cos x e ^x – ∫ [ ( – Sin x ) e ^x ] dx
= e ^x ( Sin x – Cos x ) – ∫ Sin xe ^ x dx
i = e ^x ( sin x – cos x ) – i + C | Square ∫ Sin xe ^ x = I |
2 I = e ^x ( sin x – Cos x ) + C | C is an arbitrary constant |
i = e^x ( sin x – cos x ) / 2 + C
( 3 ) ∫ ( ( lnx ) / ( 1 + ln x ) ² ) dx
= ∫( ( 1 + ln x – 1 ) / ( 1 + ln x ) ² dx
= ∫ ( dx / ( 1 + ln x ) ) – ∫ ( 1 / ( 1 + ln x ) ² ) ) dx | Apply integration by parts |
= ( 1 / ( 1 + ln x )) ∫dx – ∫ [ d / dx ( 1 / ( 1 + ln x ) ∫ dx ] dx
– ∫[ 1 / ( 1 + ln x ) ² ] dx+C
| C is an arbitrary constant |
= x / ( 1 + ln x ) – ∫[ ( – 1 ) / ( 1 + ln x ) ² × ( 1 / x ) × ( x ) ] dx
– ∫ ( dx /( 1+ lnx ) ² +
against
= x / ( 1 + ln x ) + ∫( dx / ( 1 + ln x ) ² – ∫ dx / ( 1 + ln x ) ² + C
= x / ( 1 + ln x ) + C
Now try the following:
∫ Cos 2x ln ( 1 + tan x ) dx
After going through the worked examples above, integration by parts may not be your nightmare anymore. If you still have any trouble solving, the help of an e-tutor is suggested. You can login to [http://www.learningexpress.biz] or you can send email
to [email protected]. Learning Express can provide you with the best math online
tutores A session on online integration would be of great help in case of practicing solving integration online in an interactive way.
