To determine Solve the integral of the given vector expression. Explanation Given information: Consider the given vector function. r ( t ) = sec t i + tan 2 t j − t sin t k (1) Calculation: Evaluate the integral of the vector function in Equation (1) with limits from 0 to π 4 . ∫ 0 π / 4 r ( t ) d t = ∫ 0 π / 4 ( sec t i + tan 2 t j − t sin t k ) d t = ∫ 0 π / 4 ( sec t ) d t i + ∫ 0 π / 4 ( tan 2 t ) d t j − ∫ 0 π / 4 ( t sin t ) d t k ∫ 0 π / 4 r ( t ) d t = ∫ 0 π / 4 ( sec t ) d t i + ∫ 0 π / 4 ( 1 − sec 2 t ) d t j − ∫ 0 π / 4 ( t sin t ) d t k { ∵ tan 2 t = 1 − sec 2 t } (2) Calculate the integral of t sin t . x = ∫ t sin t d t (3) Consider the expressions integration by parts. u = t and v ′ = sin t . Therefore, u ′ = d t and v = − cos t . Consider the general expression to find the integration. x = u v − ∫ v d u Substitute t for u , − cos t for v , and dt for du . x = t ( − cos t ) − ∫ ( − cos t ) d t x = − t cos t + sin t (4) Compare Equations (3) and (4)

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ISBN: 9780134439099

Author: Hass, Joel., Heil, Christopher , WEIR, Maurice D.

Publisher: Pearson,

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Chapter 13.2, Problem 10E

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Solve the integral of the given vector expression.

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Thomas' Calculus

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