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Calculus mastery assignment for module one


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Module 1 Mastery Assignment:

Module 1 Mastery Assignment By: Joseph Lutz

Integration by Partial Fractions :

Integration by Partial Fractions Integration by partial fractions is the process of integrating an expression by first breaking up the expression to express it as a sum or difference of multiple fractions By default, it is a sum of multiple fractions; however if B in the formula below becomes negative, it becomes a difference of those fractions Formula:  

Example 1: ∫1▒〖1/((x-2)(x+5)) dx〗 :

Example 1:   Rewrite the integral as the sum of two fractions Attain a common denominator for all fractions Remove denominator and find the zeros for x Plug in x-values to solve for A and B   Plug in A and B and factor out constants Integrate fractions Condense into a single natural log  

Improper Integrals :

Improper Integrals An improper integral is a definite integral with at least one limit of integration that is infinite or where an infinite limit is contained between or at the limits of integration Use L’Hopital’s Rule (must show that you are using it in proof) to solve (uses the indeterminate form) Can either converge (has a defined value) or diverge (has no defined value)

Example 2: ∫2_1^∞▒1/x dx :

Example 2:   Step 1: Apply a limit to assign t as a variable for infinity Step 2: Integrate Step 3: (Divergent) Evaluate limits of integration and evaluate the limit (since the answer is infinity, it diverges)  

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