Explanation Given information: J n = ∫ d x ( x 2 + 1 ) n Formula used: Use integration by parts formula. ∫ u d v = u v − ∫ v d u Calculation: Consider the following function. J n = ∫ d x ( x 2 + 1 ) n ........ ( 1 ) Substitute n + 1 in place of n , the function J n + 1 will be, J n = ∫ d x ( x 2 + 1 ) n + 1 ........ ( 2 ) Use integration by parts formula. ∫ u d v = u v − ∫ v d u Take u = 1 ( x 2 + 1 ) n and d v = 1 ⋅ d x and get, Differentiate u = 1 ( x 2 + 1 ) n with respect to x . d u d x = d d x [ ( x 2 + 1 ) − n ] = ( − n ) ( x 2 + 1 ) − n − 1 × d ( x 2 + 1 ) d x = − n ( x 2 + 1 ) n + 1 × ( 2 x + 0 ) = − 2 n x ( x 2 + 1 ) n + 1 The value of v will be v = ∫ d v = ∫ 1 d x = x Substitute in integral ( 1 ) , J n = ∫ 1 ( x 2 + 1 ) n × 1 d x = 1 ( x 2 + 1 ) n × x − ∫ x × ( − 2 n x ( x 2 + 1 ) n + 1 ) d x = x ( x 2 + 1 ) n + 2 n ∫ x 2 ( x 2 + 1 ) n + 1 d x = x ( x 2 + 1 ) n + 2 n ∫ ( x 2 + 1 ) − 1 ( x 2 + 1 ) n + 1 d x Simplify further, J n = x ( x 2 + 1 ) n + 2 n ∫ [ ( x 2 + 1 ) ( x 2 + 1 ) n + 1 − 1 ( x 2 + 1 ) n + 1 ] d x =

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ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.3, Problem 54E

To determine

To prove

Jn+1=(1−12n)Jn+(12n)x( x 2 +1)n

Then use this recursion relation to calculate J2 and J3.

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Chapter 8.3, Problem 53E

Chapter 8.3, Problem 55E

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To prove J n + 1 = ( 1 − 1 2 n ) J n + ( 1 2 n ) x ( x 2 + 1 ) n Then use this recursion relation to calculate J 2 and J 3 .

Solution Summary: The author explains how to use a recursion relation to calculate J_n+1=(1-12n)

Videos

To prove

Jn+1=(1−12n)Jn+(12n)x( x 2 +1)n

Then use this recursion relation to calculate J2 and J3.

Want to see the full answer?

Chapter 8 Solutions

To prove J n + 1 = ( 1 − 1 2 n ) J n + ( 1 2 n ) x ( x 2 + 1 ) n Then use this recursion relation to calculate J 2 and J 3 .

Solution Summary: The author explains how to use a recursion relation to calculate J_n+1=(1-12n)

Videos

To prove Jn+1=(1−12n)Jn+(12n)x( x 2 +1)n Then use this recursion relation to calculate J2 and J3.

Want to see the full answer?

Chapter 8 Solutions

To prove

Jn+1=(1−12n)Jn+(12n)x( x 2 +1)n

Then use this recursion relation to calculate J2 and J3.