Question

Integrate
$\displaystyle \int\dfrac{\ln(x)}{x^3} dx$

Answer 1

Step-by-step explanation:

We have,

$\displaystyle \int\dfrac{\ln(x)}{x^3} dx$

$\displaystyle = \ln(x) \int\dfrac{dx}{x^3} - \int \left[ \dfrac{d}{dx}( \ln(x)) \int \dfrac{dx}{ {x}^{3} }\right]dx$

$\displaystyle = \ln(x) \cdot\dfrac{x ^{ - 2} }{ - 2} - \int \left[ \dfrac{1}{x} \cdot \dfrac{x ^{ - 2} }{ - 2 }\right]dx$

$\displaystyle = - \dfrac{ \ln(x) }{ 2 {x}^{2} } + \dfrac{1}{2} \int \left[ \dfrac{1}{x} \cdot \dfrac{1 }{ {x}^{2} }\right]dx$

$\displaystyle = - \dfrac{ \ln(x) }{ 2 {x}^{2} } + \dfrac{1}{2} \int \dfrac{dx}{{x}^{3} }$

$\displaystyle = - \dfrac{ \ln(x) }{ 2 {x}^{2} } + \dfrac{1}{2} \dfrac{x^{ - 2} }{ - 2 } + c$

$\displaystyle = - \dfrac{ \ln(x) }{ 2 {x}^{2} } -\dfrac{1}{ 4 {x}^{2} } + c$

Answer 2

Solution:-

$\implies \int \: \frac{ ln(x) }{x {}^{3} }$

we know that formula of

$\implies \pink{ \int \: f(x). \: g(x) \: \: dx = f(x) \int \: g(x) - \int( \frac{d}{dx}f(x ) \int \: g(x)}$

$\implies \: ln(x) \int \: \frac{1}{x {}^{3} } - \int \: ( \frac{d}{dx} ln(x) \int \: \frac{1}{x {}^{3} } \\ \\ \implies \: ln(x) \frac{x {}^{ - 2} }{ - 2} - \int \: \frac{1}{x}.x {}^{ - 2} \frac{1}{ - 2} \\ \\ \implies \: ln(x) \frac{1}{ - 2x {}^{2} } + \frac{1}{2} \int \: \frac{1}{x {}^{3} }$

$\implies \: ln(x) \frac{1}{ - 2x {}^{2} } + \frac{1}{2} \frac{x {}^{ - 2} }{ - 2} \\ \\ \implies \: - \frac{1}{2} ln(x) \frac{1}{x {}^{2} } - \frac{1}{4} x {}^{ - 2} + c$

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thanks