Directions: In this project a computer algebra system (mainly Mathematica) is used to investigate integrals of families of functions. By observing the patterns that occur in the integrals of several remembers of the family, you will first guess and then prove, a general formula for the integrals of a member of the family. You will turn in all the commands (formulas used) in Mathematica or otherwise and the results you obtained for all of the items.

1. (a) Use Mathematica to evaluate the following integrals. (2 points each)

i) ii)

iii) iv)(b) Based on the pattern of your responses in part (a),

**guess the value of the integral.**(2 points), if . What if(c) Check your guess by a using computer algebra system (CAS)/Mathematica to evaluate the integral in part (b).

**Then prove it using partial fractions**. (4 points)(d) Evaluate the definite integral (Simplify your answer, no need for Mathematica!) (3 points)

(e) Evaluate the definite integral using Mathematica as well as using partial fraction.(3 points)

(f)Plot the rational function f(x)= within the interval of [2,5] using Mathematica. (1 point)(g)Use

**Trapezoidal and Simpson’s Rules**with*n=10*to approximate the integral in the interval [2,5] round to the nearest ten-thousandth and calculate the error with the result obtained in (e). (4 points)dx

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