x -log^2 x+ x^2 x^3 dx , ,is , (where C is constant of integratio… - Vidyadip

MCQ

$\int {\frac{{\log x -log^2\ x+ x^2}}{{{x^3}}}} dx\,\,is\, $ (where $C$ is constant of integration)

A
$\frac{{\log x + 2x\,\log x}}{{2{x^2}}} + C$
B
$log^2x\, +\,2xlogx\,+\,C$
✓
$\frac{{{{\log }^2}x + 2{x^2}\,\log x}}{{2{x^2}}} + C$
D
$\frac{{\log x + 2{x^2}\,\log x}}{{2{x^2}}} + C$

✓

Answer

Correct option: C.

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

c
$\int \frac{\log x}{x}\left(\frac{1-\log x}{x^{2}}\right) d x+\int \frac{d x}{x}$

Put $\frac{\log x}{x}=t \Rightarrow \int t d t+\ln x+c$

$\Rightarrow \frac{1}{2}\left(\frac{\log x}{x}\right)^{2}+\log x+c$

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