Explanation Formula used: ∫ u n e a u d u = u n e a u a − n a ∫ u n − 1 e a u d u Calculation: The given integral is ∫ x 3 e − x d x . Let u n e a u = u 3 e − u . Then the integral can be rewritten as follows. ∫ x 3 e − x d x = ∫ u 3 e − u d u The integral ∫ u 3 e − u d u is similar to ∫ u n e a u d u , where a = − 1 , u = x , n = 3 . By using the above formula, the given integral can be evaluated as follows. ∫ x 3 e − x d x = u n e a u a − n a ∫ u n − 1 e a u d x = x 3 e ( − 1 ) ( x ) − 1 − 3 − 1 ∫ x 3 − 1 e ( − 1 ) ( x ) d x = x 3 e − x − 1 − 3 − 1 ∫ x 2 e − x d x (1) Again the integral is ∫ x 2 e − x d x . Let u n e a u = u 2 e − u . Then the integral can be rewritten as follows. ∫ x 2 e − x d x = ∫ u 2 e − u d u The integral ∫ u 2 e − u d u is similar to ∫ u n e a u d u , where a = − 1 , u = x , n = 2 . By using the above formula, the given integral can be evaluated as follows. ∫ x 2 e − x d x = u n e a u a − n a ∫ u n − 1 e a u d x = x 2 e ( − 1 ) ( x ) − 1 − 2 − 1 ∫ x 2 − 1 e ( − 1 ) ( x ) d x = x 2 e − x − 1 − 2 − 1 ∫ x e − x d x Again the integral is ∫ x e − x d x

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Author: Raymond Barnett, Michael Ziegler

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Chapter 13.4, Problem 53E

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To find: The value of the integral ∫x3e−xdx.

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Chapter 13.4, Problem 52E

Chapter 13.4, Problem 54E

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Ch. 13.1 - Prob. 1MP Ch. 13.1 - Prob. 2MP Ch. 13.1 - Prob. 3MP Ch. 13.1 - Prob. 4MP Ch. 13.1 - Matched Problem 5 Find the area bounded by...Ch. 13.1 - Prob. 6MP Ch. 13.1 - Prob. 7MP Ch. 13.1 - Prob. 8MP Ch. 13.1 - Prob. 1ED Ch. 13.1 - Prob. 1E

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The value of the integral ∫ x 3 e − x d x .

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To find: The value of the integral ∫x3e−xdx.

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