Explanation Given: The expression is ∫ 9 x 2 e − x d x . Formula used: Write the integration by part formula: ∫ u d v = u v − ∫ v d u Calculation: Consider the following expression: ∫ 9 x 2 e − x d x Assume u = 9 x 2 . Determine d u by take a derivative: d u = 18 x d x Assume d v = e − x d x . Determine v by taking an anti-derivative. ∫ d v = ∫ e − x d x v = − e − x Recall the integration by part formula: ∫ u d v = u v − ∫ v d u Substitute 9 x 2 for u , e − x d x for d v , − e − x for v and 18 x d x for d u . ∫ 9 x 2 e − x d x = 9 x 2 ( − e − x ) − ∫ − e − x 18 x d x = − 9 x 2 e 2 x + ∫ 18 x e − x d x …… (1) Use the integration by parts a second time. Now, consider the expression: ∫ 18 x e − x d x Assume U = 18 x . Determine d U by take a derivative: d U = 18 d x Assume d V = e − x d x . Determine V by taking an anti-derivative

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Chapter 13, Problem 32RE

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To calculate: The value of the expression ∫9x2e−xdx by using the integration by parts twice.

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Chapter 13, Problem 31RE

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The value of the expression ∫ 9 x 2 e − x d x by using the integration by parts twice.

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To calculate: The value of the expression ∫9x2e−xdx by using the integration by parts twice.

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