Calculus trapezoidal and simspsons rule using python programming

Question

this question is to be sovled using python programming

For each of the integral given below, do the following using Python Programming

a)         plot the graph related to Trapezoidal's rule or Simpson's Rule for number of intervals that have chosen from N = 5 to 20 intervals (choose one only)

b)         approximate each of the given integral using both Trapezoidal's rule and Simpson's rule using two different number of intervals for each case

1) ∫_0^2▒〖"(" x^3 "+x + 1 dx" 〗

2)∫_0^1▒〖sin⁡ (x^2)" dx" 〗

3)∫_2^4▒〖" " 1/(1+x^3 )〗 " dx"

4)∫_2^6▒〖" " 1/(ln x)〗 " dx"

Answer 1

1) Trapezoidal: n=10 --> 8.04

n=20 ---> 8.01

Simpsons: n=10 ---> 8

n=20 ---> 8

the exact answer is (1/4)x^4 + (1/2)x^2 + x , x=2

= 4 + 2 + 2 = 8

2) cannot be integrated without special functions

Trapezoidal: n=10 ---> 0.311170811217031

n=20 ---> 0.310493552903314

Simpsons: n=10 ---> 0.310268

n=20 ---> 0.310268

3) the anti-derivative is

[ln(x^2-x+1) - 2 ln |x+1| + sqrt(3) arctan((2x-1)/sqrt(3))]/6+c

integration steps:

a) factors denomaintor via SOAP formula: (x+1)(x^2-x+1)

b) partial fraction decomposition gets you:

(1/3)(1 / x+1) - (1/3) [ (x-2)/(x^2-x+1)]

c) first term integrates to ln|x+1|

d) second term: rewrites x-2 = (1/2)(2x-1) -3/2

e) splits the second integral into these two pieces, the first

can be integrated via subsituteion U=x^2-x+1

f) the second is done by completing of the square: x^2-x+1 = (x-1/2)^2 + 3/4

and then letting u= (2x-1)/sqrt(3)

The exact answer is 0.088141272

Trapezoidal: n=10--> 0.0885962203610578

n=20 --> 0.0882551985818365

Simpsons: n=10 ---> 0.0881415

n=20 ----> 0.0881413

4) cannot be integrated without special functions

Trapezoidal: n=10---> 3.190042579714

n=20 ---> 3.18034139625903

Simpsons: n = 10 ---> 3.17711

n=20 --->3.17706