Adaptive Integration

Consider the function y =

Sinx . Evaluate x



 0

Sinx Sinx . dx and find the area of y = x x

Sinx is a divergent improper integral; for this reason, it is good to use adaptive integration x technique. It is known that an improper integral is easy to calculate away from its singularity or bad point, and it is useful to use lots of points to approximate the integral near its singularity Sinx but not so many elsewhere. The graph of y = is the following: x

Y=

1

0.8

0.6

0.4

0.2

10

First, we will estimate from 0 to 1 of y =

20

30

40

50

Sinx . After that we will calculate the derivative of y = x

Sinx and will estimate the area from 1 to 50. x Cosx converges more fast; so, if we estimate area from 1 to 50 of the x^2 50 1 Cosx Sinx function y=  dx , and the add the area with the area of the function y =  dx , x 1 x^2 0  Sinx we wil get an approximate area for the function y =  dx . x 0

The function y =

The graph of y =

Cosx is the following: x^2

1

0.8

0.6

0.4

0.2

10

20

30

40

50

1

Sinx dx is 0.946083. x 0 50 Cosx The area of y =  dx is -0.0845312+1 1 x^2 So, the total area is approximately = 1.86155. The area of y =



(0.946083+1-0.0845312) 50

The derivation of the numbers from the integral

Cosx

 x ^ 2 dx is the following: 1

u

1 x

du =

v  Cosx dv = Sinx dx

 du x^2

The partial derivative formula is: u*v -  vdu . Plugging the values yield: (

Cosx | x = 0 to  ) - ( x

50

Cosx

 x^2

dx ) = 1 - -0.0845312.

1

The answer 1.86 is much accurate because if we use NIntegrate, we get the answer 1.87.