Integration becomes very handy if you choose to use Wolfram Mathematica. You can evaluate definite and indefinite integrals using two ways.

**1.**You may use "

*Integrate[f, x]*" command for the indefinite integrals, and "

*Integrate[f, {x, upper limit, lower limit}]*" for the definite integrals.

**2.**Use int to enter

**∫**and then use to enter the lower limit, then for the upper limit:

For example, if you would like to find out the following integral of a function using both approaches,

Now, let's look at another example, where you will find detailed Mathematica codes to determine the integral of the following:

Note that the following Mathematica program also implements a numerical integration scheme by using a built-in "

*NIntregate*" command. Look carefully the systaxes used in the following codes to display the results, you may represent them in your own fashion.

**Mathematica Codes:**

**(*Showing Exact Integration of First Problem*)**

**Print["Exact solution of \!\(\*SubsuperscriptBox[\(\[Integral]\), \**

**\(-1\), \(1\)]\)(2\!\(\*SuperscriptBox[\(x\), \(2\)]\)+3)\**

**\[DifferentialD]x"]**

**FullSimplify[\!\(**

**\*SubsuperscriptBox[\(\[Integral]\), \(-1\), \(1\)]\(\((2**

**\*SuperscriptBox[\(x\), \(2\)] + 3)\) \[DifferentialD]x\)\)]**

**(*Showing Numerical Integration of First Problem*)**

**Print["Numerical Integration by Approximation: \**

**\!\(\*SubsuperscriptBox[\(\[Integral]\), \(-1\), \**

**\(1\)]\)(2\!\(\*SuperscriptBox[\(x\), \(2\)]\)+3)\[DifferentialD]x**

**"]**

**NIntegrate[(2 x^2 + 3), {x, -1, 1}]**

**Print["Numerical Integration by Gaussian Rule Order-1: \**

**\!\(\*SubsuperscriptBox[\(\[Integral]\), \(-1\), \**

**\(1\)]\)(2\!\(\*SuperscriptBox[\(x\), \(2\)]\)+3)\[DifferentialD]x**

**"]**

**a = 2 x^2 + 3 /. x -> -Sqrt[1/(3)];**

**b = 2 x^2 + 3 /. x -> Sqrt[1/(3)];**

**c = N[a + b]**

**Program Output:**

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