Short Answer

Solve the integral: $\int x \ln \sqrt{x} d x$

The required answer is $\frac{x^{2}}{4} \ln (x) - \frac{x^{2}}{8} + c$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

We have given integral is $\int x \ln \sqrt{x} d x$ .

Step 2. Solve the integration by parts .

We have,

$\int x \ln \sqrt{x} d x = \int (\frac{x}{2} \ln x) d x = \frac{1}{2} \int x \ln x d x$

$u = \ln x d u = \frac{d x}{x}$

and

$d v = x d x v = \int x d x v = \frac{x^{2}}{2}$

The formula of integration by parts is $\int u d v = u v - \int v d u$ .

localid="1648797136253" $\frac{1}{2} \int x \ln x d x = \frac{1}{2} (\ln (x) \frac{x^{2}}{2} - \int \frac{x^{2}}{2} \frac{1}{x} d x) = \frac{1}{2} (\frac{x^{2}}{2} \ln (x) - \frac{1}{2} \int x d x) = \frac{x^{2}}{4} \ln (x) - \frac{x^{2}}{8} + c$

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Calculus

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Short Answer

Solve the integral: $\int x \ln \sqrt{x} d x$

Step by Step Solution

Step 1. Given information.

Step 2. Solve the integration by parts .

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Solve the integral: $\int x \ln \sqrt{x} d x$