Explanation Given: The given integral is ∫ 0 x 2 sin t d t . Concept used: Formula to evaluate the integral of the form ∫ sin x d x is shown below. ∫ sin x d x = − cos x + C Formula used: Fundamental theorem of Calculus I: ∫ a b f ( x ) d x = F ( b ) − F ( a ) Where F ′ ( x ) = f ( x ) Fundamental theorem of Calculus II: If f ( t ) be any continuous function over the interval [ a , b ] and the definite integral is given by G ( x ) = ∫ a x f ( t ) d t for a ≤ t ≤ b Then G is an antiderivative of f ( t ) on [ a , b ] , and the derivative of G ( x ) is given by G ′ ( x ) = d d x [ ∫ a x f ( t ) d t ] = f ( x ) Calculation: Consider the integral. F ( x ) = ∫ 0 x 2 sin t d t Now consider t = z 2 On, differentiating t = z 2 d t = 2 z d z Change the limits accordingly

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 5.4, Problem 58PS

To determine

To find: The derivative of the integral ∫0x2sintdt . Also check the result by differentiating the value of integral.

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Chapter 5.4, Problem 57PS

Chapter 5.4, Problem 59PS

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Chapter 5 Solutions

Calculus

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The derivative of the integral ∫ 0 x 2 sin t d t . Also check the result by differentiating the value of integral.

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To find: The derivative of the integral ∫0x2sintdt . Also check the result by differentiating the value of integral.

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Chapter 5 Solutions