First slide

Methods of integration

Question

$Evaluate \int \frac{\log x}{(1 + \log x)^{2}} d x$

Moderate

A

$\frac{- x}{(\log x + 1)} + c$
B

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

$\frac{x}{(\log x + 1)} + c$
D

None of these

Solution

$I = \int \frac{\log x}{(1 + \log x)^{2}} d x$
$Let \log x = t . Then x = e^{t} or d x = e^{t} d t . Thus,$
$\begin{array}{l} I = \int \frac{t e^{t}}{(t + 1)^{2}} d t \\ = \int \frac{(t + 1 - 1) e^{t}}{(t + 1)^{2}} d t (a d d a n d s u b t r a c t e^{t} i n t h e n u m e r a t o r) \\ = \int e^{t} (\frac{1}{(t + 1)} - \frac{1}{(t + 1)^{2}}) d t \\ = \frac{e^{t}}{t + 1} + c = \frac{x}{(\log x + 1)} + c u \sin g t h e f o r m u l a (\int e^{x} (f (x) + f^{'} (x)) d x = e^{x} f (x) + C) \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App