MAT 156 LAB 9 Topic 1: Different substitutions to the same integral. Assignment 1 Consider the integral NiMtJSRpbnRHNiQqKiUkc2luRyIiIiUieEciIiIlJGNvc0ciIiIlInhHIiIiJSJ4Rw==. a) Solve the integral using the substitution NiMvJSJ1RyomJSRzaW5HIiIiJSJ4RyIiIg== b) Solve the integral using the substitution NiMvJSJ1RyomJSRjb3NHIiIiJSJ4RyIiIg== c) Explain why these answers are different.

Topic 2: The method of partial fractions and completing the square. Quite often we have to integrate functions that are quotients of two polynomials NiMqJi0lIlBHNiMlInhHIiIiLSUiUUc2IyUieEchIiI=. These functions are called rational functions and we can use the method of partial fractions to integrate them. Example: NiMtJSRpbnRHNiQqJiwoKiQlInhHIiIkIiIiKiYiIiMiIiIqJCUieEciIiMiIiIhIiIiIigiIiIiIiIsLCokJSJ4RyIiJSIiIiomIiIkIiIiKiQlInhHIiIkIiIiISIiKiYiIiQiIiIqJCUieEciIiMiIiIiIiIqJiIiJCIiIiUieEciIiIhIiIiIiMiIiIhIiIlInhH We introduce the expression f:=(x^3-2*x^2+7)/(x^4-3*x^3+3*x^2-3*x+2);

NiM+JSJmRyomLCgqJCklInhHIiIkIiIiIiIiKiYiIiMiIiIpJSJ4RyIiIyIiIiEiIiIiKCIiIiIiIiwsKiQpJSJ4RyIiJSIiIiIiIiomIiIkIiIiKSUieEciIiQiIiIhIiIqJiIiJCIiIiklInhHIiIjIiIiIiIiKiYiIiQiIiIlInhHIiIiISIiIiIjIiIiISIi

fpar:=convert(f, parfrac,x);

NiM+JSVmcGFyRywoKiYiIiQiIiIsJiUieEciIiIiIiIhIiIhIiIhIiIqKCIiJiEiIiwmKiYiIzgiIiIlInhHIiIiIiIiIiInIiIiIiIiLCYqJCklInhHIiIjIiIiIiIiIiIiIiIiISIiIiIiKigiIigiIiIiIiYhIiIsJiUieEciIiIiIiMhIiIhIiIiIiI=

fpar is equal to f, only it is written in a form that is easier to integrate. Now we can define the integral of fpar and ask Maple to compute it. We recognize ourselves that NiMsJiUieEciIiIiIiMhIiI= in the denominator will give NiMtJSNsbkc2IywmJSJ4RyIiIiIiIyEiIg== and the NiMsJiUieEciIiIiIiIhIiI= in the denominator will give NiMtJSNsbkc2IywmJSJ4RyIiIiIiIiEiIg==. Also we can split the expression with NiMsJiokJSJ4RyIiIyIiIiIiIiIiIg== in the denominator into NiMqJiIiJyIiIi0iIiY2IywmKiQlInhHIiIjIiIiIiIiIiIiISIi plus NiMqKCIjOCIiIiUieEciIiItIiImNiMsJiokJSJ4RyIiIyIiIiIiIiIiIiEiIg==. The first term gives an arctan (NiMlInhH), while the second gives NiMtJSNsbkc2IywmKiQlInhHIiIjIiIiIiIiIiIi with appropriate constant in front. fint:=int(fpar, x);

NiM+JSVmaW50RywqKiYiIiQiIiItJSNsbkc2IywmJSJ4RyIiIiIiIiEiIiIiIiEiIiomIyIjOCIjNSIiIi0lI2xuRzYjLCYqJCklInhHIiIjIiIiIiIiIiIiIiIiIiIiIiIiKiYjIiInIiImIiIiLSUnYXJjdGFuRzYjJSJ4RyIiIiIiIiomIyIiKCIiJiIiIi0lI2xuRzYjLCYlInhHIiIiIiIjISIiIiIiIiIi

Naturally Maple can do all these things by itself: gint:=int(f, x);

NiM+JSVnaW50RywqKiYiIiQiIiItJSNsbkc2IywmJSJ4RyIiIiIiIiEiIiIiIiEiIiomIyIjOCIjNSIiIi0lI2xuRzYjLCYqJCklInhHIiIjIiIiIiIiIiIiIiIiIiIiIiIiKiYjIiInIiImIiIiLSUnYXJjdGFuRzYjJSJ4RyIiIiIiIiomIyIiKCIiJiIiIi0lI2xuRzYjLCYlInhHIiIiIiIjISIiIiIiIiIi

Assignment Part 2: Find the partial fraction decomposition and integrate: NiMtJSRpbnRHNiQqJiwoKiQlInhHIiIpIiIiKiYiIiMiIiIlInhHIiIiIiIiIiIiISIiIiIiKiYsJiUieEciIiIiIiIhIiIiIiQsJiokJSJ4RyIiIyIiIiIiJCIiIiIiIyEiIiUieEc= .

NiMsMiomIiIjISIiJSJ4RyIiIyIiIiomIiIkIiIiJSJ4RyIiIiIiIiomIiIiIiIiKiYiIzsiIiIpLCYlInhHIiIiIiIiISIiIiIjIiIiISIiISIiKiYjIiNQIiNLIiIiLSUjbG5HNiMsJiUieEciIiIiIiIhIiIiIiIiIiIqKCIjWyEiIiwmKiYiIyEpIiIiJSJ4RyIiIiIiIiIiJyEiIiIiIiwmKiQpJSJ4RyIiIyIiIiIiIiIiJCIiIiEiIiIiIiomIyIkbigiJClHIiIiKiYiIiQjIiIiIiIjLSUnYXJjdGFuRzYjLCQqKCIiJCEiIiUieEciIiIiIiQjIiIiIiIjIiIiIiIiIiIiISIiKiYiIiIiIiIqJiIiIyIiIiwmJSJ4RyIiIiIiIiEiIiIiIiEiIiEiIiomIyIjUCIjayIiIi0lI2xuRzYjLCYqJCklInhHIiIjIiIiIiIiIiIkIiIiIiIiISIi

Sometimes this does not work because the roots of the denominator are not real numbers, or involve radicals. If the denominator does not have real roots, we can complete the square. The command is completesquare(expression, x) with(student);

NiM3QCUiREclJURpZmZHJSpEb3VibGVpbnRHJSRJbnRHJSZMaW1pdEclKExpbmVpbnRHJShQcm9kdWN0RyUkU3VtRyUqVHJpcGxlaW50RyUqY2hhbmdldmFyRyUvY29tcGxldGVzcXVhcmVHJSlkaXN0YW5jZUclJ2VxdWF0ZUclKmludGVncmFuZEclKmludGVyY2VwdEclKWludHBhcnRzRyUobGVmdGJveEclKGxlZnRzdW1HJSltYWtlcHJvY0clKm1pZGRsZWJveEclKm1pZGRsZXN1bUclKW1pZHBvaW50RyUocG93c3Vic0clKXJpZ2h0Ym94RyUpcmlnaHRzdW1HJSxzaG93dGFuZ2VudEclKHNpbXBzb25HJSZzbG9wZUclKHN1bW1hbmRHJSp0cmFwZXpvaWRH

f:=1/(x^2+6*x+14);

NiM+JSJmRyomIiIiIiIiLCgqJCklInhHIiIjIiIiIiIiKiYiIiciIiIlInhHIiIiIiIiIiM5IiIiISIi

fpar:=convert(f, parfrac, x);

NiM+JSVmcGFyRyomIiIiIiIiLCgqJCklInhHIiIjIiIiIiIiKiYiIiciIiIlInhHIiIiIiIiIiM5IiIiISIi

newf:=completesquare(f, x);

NiM+JSVuZXdmRyomIiIiIiIiLCYqJCksJiUieEciIiIiIiQiIiIiIiMiIiIiIiIiIiYiIiIhIiI=

This suggests the substitution NiMvJSJ1RywmJSJ4RyIiIiIiJCIiIg==. Assignment Part 3: Perform this substitution and compute the integral.

Assignment Part 4 : Compute the integral NiMtJSRpbnRHNiQqJiwqKiQlInhHIiIkIiIiKiYiIiYiIiIqJCUieEciIiMiIiIiIiIqJiIiKCIiIiUieEciIiIhIiIiIiIiIiIiIiIsKCokJSJ4RyIiIyIiIiUieEciIiIiIiIiIiIhIiIlInhH. using techniques we learned in this lab.