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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