First slide

Introduction to integration

Question

The value of $\int \log (1 - \sqrt{x}) d x$ is

Moderate

A

$x \log (1 - \sqrt{x}) - \frac{1}{2} x - \frac{3}{2} \sqrt{x} + C$
B

$(x - 1) \log (1 - \sqrt{x}) - \frac{1}{2} x - \sqrt{x} + C$
C

$x^{2} \log (1 - \sqrt{x}) - \frac{1}{2} \sqrt{x} + C$
D

none of these

Solution

Integrating by parts taking log $(1 - \sqrt{x})$ as the first function, the given integral can be written as
$\begin{array}{l} x \log (1 - \sqrt{x}) + \frac{1}{2} \int \frac{\sqrt{x}}{1 - \sqrt{x}} d x \\ = x \log (1 - \sqrt{x}) + \int \frac{t^{2}}{1 - t} d t \\ - x \log (1 - \sqrt{x}) + \int (- t - 1 + \frac{1}{1 - t}) d t \\ = (x - 1) \log (1 - \sqrt{x}) - \frac{x}{2} - \sqrt{x} + C . \end{array}$

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