Question

dx
The integral
Dx/x^2(x^4+1)^3/4

Answer 1

Step-by-step explanation:

We have,

$\int \frac{dx}{ {x}^{2} ( {x}^{4} + 1) ^{ \frac{3}{4} } }$

$\int \frac{dx}{ {x}^{2} . {x}^{3}(1 + \frac{1}{ {x}^{4} } )^{ \frac{3}{4} } }$

$\int \frac{dx}{ {x}^{5} (1 + \frac{1}{ {x}^{4} })^{ \frac{3}{4} } }$

$let \: \: 1 + \frac{1}{ {x}^{4} } = t \\ \implies - \frac{dx}{ {x}^{5} } = dt$

$\int \frac{ - dt}{ {t}^{ \frac{3}{4} } }$

$= - \frac{ {t}^{ - \frac{3}{4} + 1} }{ - \frac{ 3}{4} + 1 } + c$

$= - \frac{ {t}^{ \frac{1}{4} } }{ \frac{1}{4} } + c$

$= - 4 {t}^{ \frac{1}{4} } + c$

$= - 4 {(1 + \frac{1}{ {x}^{4} }) }^{ \frac{1}{4} } + c$

$= - 4 \frac{( {x}^{4} + 1) ^{ \frac{1}{4} } }{x} + c$