Question

question is in attachment​

Answer 1

we have to integrate function ,

$\bf\frac{(x+1)(x+log\ x)^2}{x}$

first of all, resolve the given function.

$\bf\frac{(x+1)(x+logx)^2}{x}$

= $\bf\left(\frac{x+1}{x}\right)(x+logx)^2$

= $\bf\left(1+\frac{1}{x}\right)(x+logx)^2$

Let x + logx = P .....(1)

differentiating both sides,

dx + 1/x . dx = dP

=> (1 + 1/x) dx = dP ........(2)

now, $\bf\int{\frac{(x+1)(x+logx)^2}{x}}\,dx=\bf\int{\left(1+\frac{1}{x}\right)(x+logx)^2}\,dx$

from equations (1) and (2),

$\bf\int{\left(1+\frac{1}{x}\right)(x+logx)^2}\,dx=\bf\int{P^2}\,dP$

= $\bf\left[\frac{P^3}{3}\right]+C$

now putting P = (x + logx)

= $\bf\left[\frac{(x+logx)^3}{3}\right]+C$

Answer 2

HOPE ITS HELP YOU..........