Explanation Given Information: The integral function is ∫ x 2 cos ( 2 x ) d x . Formula used: If u and v are two continuous functions of x then D ( u ) is the derivative of u and I ( v ) is the antiderivative of v . Calculation: Consider the integral, ∫ x 2 cos ( 2 x ) d x , To evaluate the integral ∫ x 2 cos ( 2 x ) d x , suppose that the function u is x 2 and v is cos x . Now, evaluate D ( u ) , I ( v ) and record in the table. The steps to create the table are shown below: In the table, in the column D, substitute the value of function u , and in the column I, substitute the value of v. Below x, insert the value of D ( x ) = 2 x , and in the column I, insert the value of I ( cos 2 x ) = 1 2 s i n 2 x . The arrow shows the product of two terms as 1 2 x 2 s i n 2 x . The plus sign on the left of the table is used to indicate that the product will appear with positive sign in the formula. Insert the value of D ( x ) = 2 in column D , and in column I, insert the value of I ( 1 2 sin 2 x ) = − 1 4 cos ( 2 x ) . Next arrow shows the product of D ( x ) = 2 with I ( − 1 4 cos ( 2 x ) ) = − 1 8 sin ( 2 x ) . The minus sign on the left of the table is indicating that the product will appear with a negative sign in the formula. The arrow shows the product of derivative of constant with I ( − 1 4 cos ( 2 x ) ) = − 1 8 sin ( 2 x )

7th Edition

ISBN: 9781337514309

Author: Waner

Publisher: Cengage

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Chapter 9.3, Problem 47E

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To calculate: The value of the integral ∫x2cos(2x) dx using integration by parts.

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Chapter 9.3, Problem 46E

Chapter 9.3, Problem 48E

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The value of the integral ∫ x 2 cos ( 2 x ) d x using integration by parts.

Solution Summary: The author calculates the integral's value using integration by parts.

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To calculate: The value of the integral ∫x2cos(2x) dx using integration by parts.

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Chapter 9 Solutions