Explanation Given: d d x ∫ 0 x 2 t d t t + 1 Key concepts applied: When a limit of integration is a function of variable of differentiation we need to use both chain rule and fundamental theorem of calculus in order to find derivative of integral. Say if the upper limit of integral is a function, then when finding the derivative of that integral, the upper limit replaces the variable of integration wherever it appears in the integrand and the result is multiplied by the derivative of the upper limit. The general formula is given below: d d x ∫ a g ( x ) f ( t ) ⋅ d t = f ( g ( x ) ) ⋅ g ' ( x ) Here in our problem we need to identify g ( x ) and f ( t ) . Then we need to identify the composite function f ( g ( x ) ) and derivative of the upper limit g ' ( x ) . Finally substitute those values in the formula to find the derivative of the given integral Formula used: d d x ∫ a g ( x ) f ( t ) ⋅ d t = f ( g ( x ) ) ⋅ g ' ( x ) I f g ( x ) = k ⋅ x n , t h e n g ' ( x ) = k ⋅ n ⋅ x n − 1 Calculation: Description d d x ∫ 0 x 2 t d t t + 1 Step 1: Identify g ( x ) and f ( t ) by comparison d d x ∫ a g ( x ) f ( t ) ⋅ d t ⇔ d d x ∫ 0 x 2 t t + 1 ⋅ d t g ( x

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 5.5, Problem 31E

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The derivative of integral with the given limits

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Chapter 5.5, Problem 30E

Chapter 5.5, Problem 32E

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The derivative of integral with the given limits

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Chapter 5 Solutions