First slide

Introduction to integration

Question

If $f (x) = \frac{\log x}{x^{3}}$ , then its antiderivative $F (x)$
given by

Easy

A

$C - \frac{1}{2 x^{2}} \log (x \sqrt{e})$
B

$C + \frac{1}{2 x^{2}} \log x$
C

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

$C + \frac{1}{2 x^{3}} \log (x e)$

Solution

$F (x) = \int \frac{\log x}{x^{3}} d x = \log x \int \frac{1}{x^{3}} d x -$ $\int \frac{1}{x} (\int \frac{1}{x^{3}} d x) d x$
$= - \frac{1}{2 x^{2}} \log x + \frac{1}{2} \int \frac{1}{x^{3}} d x + Constant$
$\begin{array}{l} = - \frac{1}{2 x^{2}} \log x - \frac{1}{4 x^{2}} + C \\ = - \frac{1}{2 x^{2}} [\log x + \frac{1}{2} \log e] + C \\ = - \frac{1}{2 x^{2}} \log x \sqrt{e} + C \end{array}$

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