Explanation Given: Givenintegral is ∫ 0 1 / 2 d x x ( ln x ) p Concept used: Integral test: Consider a series: ∑ n ∞ f ( x ) If ∫ n ∞ f ( x ) d x is finite value, then the series is convergent otherwise it is divergent. Calculation: ∫ 0 1 / 2 d x x ( ln x ) p = ∫ 0 1 / 2 1 ( ln x ) p × d x x Let ln x = t ⇒ 1 x d x = d t At x = 0 , t = ∞ At x = 1 / 2 , t = ln ( 1 / 2 ) = − ln 2 = ∫ ∞ − ln 2 d t ( t ) p = ∫ ∞ − ln 2 t − p d t = − ∫ − ln 2 ∞ t − p d t [ Interchanging limits

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence t…

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

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Chapter 7.7, Problem 49PS

To determine

To find the value of p , for which ∫01/2dxx (lnx) p converges and also find the value of integral.

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Chapter 7.7, Problem 48PS

Chapter 7.7, Problem 50PS

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Chapter 7 Solutions

Calculus

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To find the value of p , for which ∫ 0 1 / 2 d x x ( ln x ) p converges and also find the value of integral.

Solution Summary: The author explains how to find the value of p and integral. If the series is convergent, otherwise it is divergent.

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To find the value of p , for which ∫01/2dxx (lnx) p converges and also find the value of integral.

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Chapter 7 Solutions