Explanation Given: • Equation: ∫ x n ⋅ e x ⋅ d x = P n ( x ) ⋅ e x + C • P n ( x ) = x n − n P n − 1 ( x ) • P 0 ( x ) = 1 Key Concepts Used: Here in this problem, first we have to find the definition of P n ( x ) from the given equation: ∫ x n ⋅ e x ⋅ d x = P n ( x ) ⋅ e x + C ; then we have to prove that P n ( x ) = x n − n P n − 1 ( x ) and in final we have to find the value of P n ( x ) for n = 1 , 2 , 3 , 4 . To define P n ( x ) , we can use mathematical induction. For that first we have to find P 0 ( x ) . Then we should prove that P 1 ( x ) is true. Then we have to prove P n + 1 ( x ) is true if P n ( x ) true. By means of which we have to find the value of P n ( x ) . In final we have to substitute n = 1 , 2 , 3 , 4 in P n ( x ) to find the corresponding values. Calculation: Description: ∫ x n ⋅ e x ⋅ d x = P n ( x ) ⋅ e x + C Step 1 : Write the given equation. ∫ x n ⋅ e x ⋅ d x = P n ( x ) ⋅ e x + C Step 2 : We can use mathematical induction to define and prove the given equations. First we have to find P 0 ( x ) . For that substitute n = 0 It is also given that P 0 ( x ) = 1 . ∫ x n ⋅ e x ⋅ d x = P n ( x ) ⋅ e x + C ⇒ P n ( x ) ⋅ e x + C = ∫ x n ⋅ e x ⋅ d x ⇒ P 0 ( x ) ⋅ e x = ∫ x 0 ⋅ e x ⋅ d x ⇒ P 0 ( x ) ⋅ e x = ∫ e x ⋅ d x ⇒ P 0 ( x ) ⋅ e x = e x ⇒ P 0 ( x ) = e x e x = 1 P 0 ( x ) = 1 P 0 ( x ) = x 0 − 0 Step 3 : Substitute n = 1 to prove that P 1 ( x ) is true. ∫ x n ⋅ e x ⋅ d x = P n ( x ) ⋅ e x + C ⇒ P 1 ( x ) ⋅ e x = ∫ x 1 ⋅ e x ⋅ d x ⇒ P 1 ( x ) ⋅ e x = ∫ x ⋅ e x ⋅ d x = x ⋅ e x − ∫ e x ⋅ d x ( B y I n t e g r a t i o n b y p a r t s ) ⇒ P 1 ( x ) ⋅ e x = x ⋅ e x − ( e x ) = e x ( x − 1 ) ⇒ P 1 ( x ) = e x ( x − 1 ) e x ⇒ P 1 ( x ) = x − 1 P 1 ( x ) = x 1 − 1 P 1 − 1 ( x ) Step 4 : Consider P n ( x ) is true. ∫ x n ⋅ e ⋅ x d x = P n ( x ) ⋅ e x + C P n ( x ) = x n − n P n − 1 ( x ) Step 5 : By mathematical induction if P n ( x ) is true then P n + 1 ( x ) should be true. If we prove that then the equation is true...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 8.1, Problem 92E

To determine

• The definition of $P_{n} (x)$ from the given equation

• Proof of $P_{n} (x)$ given

• Value of $P_{n} (x)$ for $n = 1, 2, 3, 4$

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Chapter 8.1, Problem 91E

Chapter 8.1, Problem 93E

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Chapter 8 Solutions

CALCULUS (CLOTH)

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• The definition of P n ( x ) from the given equation • Proof of P n ( x ) given • Value of P n ( x ) for n = 1 , 2 , 3 , 4

Solution Summary: The author explains how to determine the definition of P n (x) from the given equation.

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• The definition of $P_{n} (x)$ from the given equation

• Proof of $P_{n} (x)$ given

• Value of $P_{n} (x)$ for $n = 1, 2, 3, 4$

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Chapter 8 Solutions

• The definition of P n ( x ) from the given equation • Proof of P n ( x ) given • Value of P n ( x ) for n = 1 , 2 , 3 , 4

Solution Summary: The author explains how to determine the definition of P n (x) from the given equation.

Videos

• The definition of Pn(x) from the given equation • Proof of Pn(x) given • Value of Pn(x) for n=1, 2, 3, 4

Want to see the full answer?

Chapter 8 Solutions

• The definition of $P_{n} (x)$ from the given equation

• Proof of $P_{n} (x)$ given

• Value of $P_{n} (x)$ for $n = 1, 2, 3, 4$