Explanation Given information: The integral function is ∫ x e x 1 + e x d x . Calculation: Show the integral function as follows: I = ∫ x e x 1 + e x d x (1) Let us consider t = 1 + e x (2) Rearrange Equation (2) t = 1 + e x t 2 = 1 + e x (3) Partially differentiate both sides of the equation (3) with respect to x . 2 t d t d x = 0 + e x 2 t d t = e x d x (4) Rearrange Equation (3) t 2 = 1 + e x e x = t 2 − 1 (5) Modify Equation (5). e x = t 2 − 1 ln e x = ln ( t 2 − 1 ) x = ln ( t 2 − 1 ) Substitute ln ( t 2 − 1 ) for x , t for 1 + e x , and 2 t d t for e x d x in Equation (1) I = ∫ ln ( t 2 − 1 ) t ( 2 t d t ) = ∫ ln [ ( t + 1 ) ( t − 1 ) ] ( 2 d t ) = 2 ∫ [ ln ( t + 1 ) + ln ( t − 1 ) ] d t = 2 [ ∫ ln ( t + 1 ) d t + ∫ ln ( t − 1 ) d t ] (6) Consider I 1 and I 2 as follows: I 1 = ∫ ln ( t + 1 ) d t (7) I 2 = ∫ ln ( t − 1 ) d t (8) Substitute I 1 for 2 ∫ ln ( t + 1 ) d t and I 2 for 2 ∫ ln ( t − 1 ) d t in Equation (6). I = 2 ∫ ln ( t + 1 ) d t + 2 ∫ ln ( t − 1 ) d t I = 2 ( I 1 + I 2 ) (9) Calculate the integral function ( I 1 ) using Equation (7): I 1 = ∫ ln ( t + 1 ) d t Consider u = ln ( t + 1 ) (10) Partially differentiate both sides of the equation (10) with respect to t . d u d t = 1 ( t + 1 ) d u = 1 ( t + 1 ) d t Consider d v = d t (11) Integrate both sides of Equation (11). ∫ d v = ∫ d t v = t Apply the ‘formula for integration by parts’: ∫ u d v = u v − ∫ v d u (12) Substitute ln ( t + 1 ) for u, d t for dv , t for v and 1 ( t + 1 ) d t for du in Equation (12). ∫ ln ( t + 1 ) d t = ln ( t + 1 ) ( t ) − ∫ t ( t + 1 ) d t = ln ( t + 1 ) ( t ) − ∫ t + 1 − 1 ( t + 1 ) d t = ln ( t + 1 ) ( t ) − ∫ [ ( t + 1 ) ( t + 1 ) − 1 t + 1 ] d t = ln ( t + 1 ) ( t ) − ∫ ( 1 − 1 t + 1 ) d t ∫ ln ( t + 1 ) d t = t ln ( t + 1 ) − t + ln ( t + 1 ) + C I 1 = ( t + 1 ) ln ( t + 1 ) − t + C (13) Calculate the integral function ( I 2 ) using Equation (7): I 2 = ∫ ln ( t − 1 ) d t Consider u = ln ( t − 1 ) (14) Partially differentiate both sides of the equation (14) with respect to t

Student Solutions Manual for Stewart's Single Variable Calculus: Early Transcendentals, 8th (James Stewart Calculus)

8th Edition

ISBN: 9781305272422

Author: James Stewart, Jeffrey A. Cole, Daniel Drucker, Daniel Anderson

Publisher: Cengage Learning

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Chapter 7.5, Problem 77E

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To evaluate: The integral function ∫xex1+exdx.

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Chapter 7.5, Problem 76E

Chapter 7.5, Problem 78E

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Student Solutions Manual for Stewart's Single Variable Calculus: Early Transcendentals, 8th (James Stewart Calculus)

Ch. 7.1 - Evaluate the integral using integration by parts...Ch. 7.1 - Prob. 2E Ch. 7.1 - Evaluate the integral. 3. xcos5xdx Ch. 7.1 - Evaluate the integral. 4. ye0.2ydy Ch. 7.1 - Evaluate the integral. 5. te3tdt Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E Ch. 7.1 - Evaluate the integral. 8. t2sintdt Ch. 7.1 - Evaluate the integral. 9. cos1xdx Ch. 7.1 - Prob. 10E

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The integral function ∫ x e x 1 + e x d x .

Solution Summary: The author explains how to evaluate the integral function, which is 'displaystyle int'.

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To evaluate: The integral function ∫xex1+exdx.

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Chapter 7 Solutions