Explanation Given: The integral h ( x ) = ∫ x x 3 cos ( t 2 ) d t . Theorem used: The Fundamental Theorem of Calculus: Suppose f is continuous on [ a , b ] , then the function g defined by g ( x ) = ∫ a x f ( t ) d t for a ≤ x ≤ b is continuous on [ a , b ] and differentiable on ( a , b ) , and g ′ ( x ) = f ( x ) . Property used: ∫ a c f ( x ) d x + ∫ c b f ( x ) d x = ∫ a b f ( x ) d x . Calculation: Let choose the change of variable functions u = x and v = x 3 . Hence d u d x = 1 2 x and d v d x = 3 x 2 . Then, the integral ∫ x x 3 cos ( t 2 ) d t can rewrite as ∫ u v cos ( t 2 ) d t . Observe that f ( t ) = cos ( t 2 ) is a trigonometric function and it is continuous everywhere, so it is continuous over any closed interval [ a , b ]

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Chapter 4.3, Problem 55E

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To find: The derivative of the function h(x)=∫xx3cos(t2) dt.

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Chapter 4.3, Problem 54E

Chapter 4.3, Problem 56E

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The derivative of the function h ( x ) = ∫ x x 3 cos ( t 2 ) d t .

Solution Summary: The author explains how the derivative of the function h(x) is based on the fundamental theorem of calculus.

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