Question: "Use the Change of Variables Formula to evaluate..."

Phoebe

Answered question

2021-11-07

Use the Change of Variables Formula to evaluate the definite integral.

\int_{1}^{e} \frac{\ln x}{x} d x

Alara Mccarthy

Skilled2021-11-08Added 85 answers

Step 1
Given integral,

\int_{1}^{e} \frac{\ln x}{x} d x

using the change of variable we have to evaluate the definite integral
Step 2
here,

\int_{1}^{e} \frac{\ln x}{x} d x

let

u = \ln x

\Rightarrow \frac{d u}{d x} = \frac{1}{x} \Rightarrow d u = \frac{1}{x} d x

x = 1, u = \ln (1) = 0

x = e, u = \ln (e) = 1

so,

\int_{1}^{e} \frac{\ln x}{x} d x = \int_{0}^{1} u d u

= {[\frac{u^{2}}{2}]}_{0}^{1} = \frac{1}{2} [1^{2} - 0] = \frac{1}{2}

\Rightarrow \int_{1}^{e} \frac{\ln x}{x} d x = \int_{0}^{1} u d u = \frac{1}{2}

this is the required answer.

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