2sin x -x+ln x=0

draw the graph
http://in.geocities. com/myisland8132/Boo k2.pub
we see that
there are roots between [0.1,1] and [2,3] use netwon's method to find the roots
let f(x)=2sin x -x+ln x
df(x)/dx=2cosx-1+1/x
x(n+1)=x(n)-f(x(n))/ f'(x(n))
x(n+1)=x(n)-[2sin x -x+ln x]/[2cosx-1+1/x]
let x(0)=0.1







interation
x(n)
x(n+1)

0
0.1
0.300447369

1
0.300447
0.515376023

2
0.515376
0.587193636

3
0.587194
0.592067109

4
0.592067
0.59208731

5
0.592087
0.59208731

6
0.592087
0.59208731

7
0.592087
0.59208731

8
0.592087
0.59208731

9
0.592087
0.59208731

10
0.592087
0.59208731

11
0.592087
0.59208731

12
0.592087
0.59208731

13
0.592087
0.59208731

14
0.592087
0.59208731

15
0.592087
0.59208731

16
0.592087
0.59208731

17
0.592087
0.59208731

18
0.592087
0.59208731

19
0.592087
0.59208731

20
0.592087
0.59208731

21
0.592087
0.59208731

22
0.592087
0.59208731

23
0.592087
0.59208731

24
0.592087
0.59208731

25
0.592087
0.59208731
let x(0)=2







interation
x(n)
x(n+1)

0
2
2.38410603

1
2.384106
2.314731096

2
2.314731
2.312744888

3
2.312745
2.312743182

4
2.312743
2.312743182

5
2.312743
2.312743182

6
2.312743
2.312743182

7
2.312743
2.312743182

8
2.312743
2.312743182

9
2.312743
2.312743182

10
2.312743
2.312743182

11
2.312743
2.312743182

12
2.312743
2.312743182

13
2.312743
2.312743182

14
2.312743
2.312743182

15
2.312743
2.312743182

16
2.312743
2.312743182

17
2.312743
2.312743182

18
2.312743
2.312743182

19
2.312743
2.312743182

20
2.312743
2.312743182

21
2.312743
2.312743182

22
2.312743
2.312743182

23
2.312743
2.312743182

24
2.312743
2.312743182

25
2.312743
2.312743182
so the roots of 2sin x -x+ln x=0 are 0.5921 and 2.3127

2sin x -x+ln x=0
x = 0.59211