Question
Evaluate: $\int \frac{\log x}{(1+\log x)^2} d x$
Problem:
$\int \frac{\log x}{(1+\log x)^2} d x$
adding and substracting 1 from numerator
$\begin{aligned} & \int \frac{1-1+\log x}{(1+\log x)^2} d x \\ & \int \frac{1+\log x}{(1+\log x)^2} d x-\int \frac{1}{(1+\log x)^2} d x \\ & \int \frac{1}{1+\log x} d x-\int \frac{1}{(1+\log x)^2} d x\end{aligned}$
For the integral
$\int \frac{1}{1+\log x} d x$
integrate by parts within the sum: ∫fg'=fg−∫f'g
$\begin{aligned} & f=\frac{1}{1+\log x} d x, g \prime=1 \\ & f \prime=-\frac{1}{(1+\log x)^2}, g=x \\ & =-\int \frac{1}{(1+\log x)^2} d x-\int-\frac{1}{(1+\log x)^2} d x+\frac{x}{\log (x)+1} \\ & =\frac{x}{\log (x)+1}\end{aligned}$
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Solve the following LPP by using graphical method.
Maximize : $Z =6 x+4 y$
Subject to $x \leq 2, x+y \leq 3,-2 x+y \leq 1, x \geq 0, y \geq 0$.
Also find maximum value of $Z$.