First slide

Introduction to integration

Question

If $\int x \log (\frac{x + 1}{x}) d x = f (x) \log (x + 1) +$ $g (x) x^{2} + L x + C$ then

Moderate

A

$f (x) = (1 / 2) (x^{2} - 1)$
B

$g (x) = \log x$
C

$L = 1$
D

none of these

Solution

$The given integral equals$
$\begin{array}{l} \int x \log (\frac{x + 1}{x}) d x = \int x \log (x + 1) d x - \int x \log x d x \\ = \frac{x^{2}}{2} \log (x + 1) - \frac{1}{2} \int \frac{x^{2}}{x + 1} d x - \frac{x^{2}}{2} \log x + \frac{1}{2} \int \frac{x^{2}}{x} d x \\ = \frac{x^{2}}{2} \log (x + 1) - \frac{x^{2}}{2} \log x \\ - \frac{1}{2} \int (x - 1 + \frac{1}{x + 1}) d x + \frac{1}{2} \int x d x \end{array}$
$\begin{aligned} = \frac{x^{2}}{2} \log (x + 1) - \frac{x^{2}}{2} \log x + \frac{x}{2} - \frac{1}{2} \log (x + 1) + C \\ \Rightarrow f (x) = \frac{x^{2}}{2} - \frac{1}{2}, g (x) = - \frac{1}{2} \log x and L = \frac{1}{2} . \end{aligned}$

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