Explanation Given: The definite integral ∫ 0 3 4 x 2 e x d x . Formula used: Write the formula for integral of f ( x ) = x n provided n ≠ − 1 . ∫ x n d x = x n + 1 n + 1 + C Write the formula for integral of f ( x ) = e x . ∫ e x d x = e x + C In the above formulae, C is the integration constant. The use of integration constant C is done only for indefinite integrals. Write the formula for derivative of f ( x ) = x n . d d x ( x n ) = n x n − 1 Write the formula for derivative of f ( x ) = e x . d d x ( e x ) = e x The integration by parts for functions u and v is given by the formula: ∫ u d v = u v − ∫ v d u The definite integral of a function f ′ ( x ) from x = a to x = b is calculated by the formula: ∫ a b f ′ ( x ) d x = | f ( x ) | a b = f ( b ) − f ( a ) Calculation: Consider the given definite integral: I = ∫ 0 3 4 x 2 e x d x Use the integration by parts formula to solve for I where u is 4 x 2 and dv is e x d x . Therefore, d u = 8 x d x and v = e x . Write the integration by parts formula. ∫ u d v = u v − ∫ v d u Substitute 4 x 2 for u , e x d x for dv, 8 x d x for du and e x for v . ∫ 0 3 4 x 2 e x d x = | 4 x 2 e x | 0 3 − ∫ 0 3 e x ( 8 x d x ) …… (1) Here, use the integration by parts further to integrate ∫ 0 3 e x ( 8 x d x ) . Write the integration by parts formula. ∫ u d v = u v − ∫ v d u Substitute 8 x for u , e x d x for dv, 8 d x for du and e x for v

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Author: Margaret Lial, Thomas Hungerford

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Chapter 13, Problem 55RE

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To calculate: The value of the definite integral given below:

∫034x2exdx

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Chapter 13, Problem 53RE

Chapter 13, Problem 56RE

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The value of the definite integral given below: ∫ 0 3 4 x 2 e x d x

Solution Summary: The author explains how to calculate the value of the definite integral, which is 20e3-8.

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To calculate: The value of the definite integral given below:

∫034x2exdx

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Chapter 13 Solutions

The value of the definite integral given below: ∫ 0 3 4 x 2 e x d x

Solution Summary: The author explains how to calculate the value of the definite integral, which is 20e3-8.

Concept explainers

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To calculate: The value of the definite integral given below: ∫034x2exdx

Want to see the full answer?

Chapter 13 Solutions

To calculate: The value of the definite integral given below:

∫034x2exdx