Explanation Given Information: The integral is ∫ 0 π x 2 sin 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 provided integral, ∫ 0 π x 2 sin x d x , To evaluate the integral ∫ 0 π x 2 sin x d x , suppose that the function u is x 2 and v is sin 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 2 , insert the value of D ( x ) = 2 x , and in the column I, insert the value of I ( sin x ) = − cos x . The arrow shows the product of two terms as − x 2 cos 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 the column I, insert the value of I ( − cos x ) = − sin x . Next arrow shows the product of D ( x ) = 2 with I ( − sin x ) = cos 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 next arrow shows the product of derivative of constant with I ( − sin x ) = cos x . The positive sign on the left of the table is indicating that the product is added to the expression. D I + x 2 sin x − 2 x − cos x + 2 − sin x − ∫ 0 cos x Now, read the values from the table and represent as: ∫ 0 π x 2 sin x d x = [ − x 2 cos x + 2 x sin x + 2 cos x ] 0 π Substitute x = π in ∫ 0 π x 2 sin x d x = [ − x 2 cos x + 2 x sin x + 2 cos x ] 0 π to determine the value of integral at upper limit

7th Edition

ISBN: 9781337291248

Author: Waner, Stefan.

Publisher: Cengage Learning,

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Chapter 9.3, Problem 51E

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Chapter 9.3, Problem 50E

Chapter 9.3, Problem 52E

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The value of the integral ∫ 0 π x 2 sin 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 ∫0πx2sinx dx using integration by parts.

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Chapter 9 Solutions