Question

Evaluate the integral. $\int_4 \frac{\ln y}{\sqrt{y}} d y$

   Evaluate the integral.
$\int_4 \frac{\ln y}{\sqrt{y}} d y$

Calculus

James Stewart 7th Edition

Chapter 7, Problem 26

Instant Answer

Solved by Expert Fuzail Shakir

07/04/2019

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Step 1: Let's identify the integral we need to evaluate: $$\int \frac{\ln y}{\sqrt{y}} dy$$ Show more…

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Evaluate the integral. $\int_4 \frac{\ln y}{\sqrt{y}} d y$

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Key Concepts

Integration Techniques

Integration is the process of finding the antiderivative or the area under the curve represented by a function. It involves various methods tailored to the form of the integrand, such as functions that are products or compositions of simpler functions.

Substitution Method

The substitution method, also known as u-substitution, is a technique used to simplify integrals by replacing a complicated expression with a new variable. This method is particularly useful when the integrand contains a composite function, as it can transform the integral into a form that is easier to evaluate.

Integration by Parts

Integration by parts is a method derived from the product rule for differentiation. It is used to integrate products of functions by selecting parts of the integrand to differentiate and integrate, turning the original integral into one or more simpler integrals.

Properties of Logarithms

Understanding the properties of logarithms, such as the power rule (e.g., ln(a^b) = b ln a), is essential when working with logarithmic functions in integration. These properties allow for the simplification and manipulation of logarithmic expressions, making them easier to integrate.

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