Question

please help me solve this

Answer 1

Solution:-

given by:-

∫ dx/x(1+x^3)^1/2

multiply by x^2 numbrator and denominetor

∫[ x^2/x^3(1+x^3)]dx

let 1+x^3 = t^2 = x^3 = t^2 -1

2x^2(dx) = 2t(dt)

x^2(dx) = t(dt)

we have,

∫( t/(t^2-1).(t^2)^1/2)dt

∫( t/(t^2-1).t)dt

∫ 1/(t^2-1)dt

$\frac{1}{2.1} log (\frac{t - 1}{t + 1} ) + c \\ \frac{1}{2} log( \frac{ \sqrt{1 + {x}^{3} } - 1 }{ \sqrt{1 + {x}^{3} } + 1 } ) + c$