The integral \int \frac{2 x^{3} - 1}{x^{4} + x} dx is equal to : (Here C is a constant of integration)

Engineering

Mathematics

Integration of Miscellaneous Functions

Question

The integral $\int \frac{2 x^{3} - 1}{x^{4} + x} dx$ is equal to :

(Here C is a constant of integration)

$\log_{e} \frac{| x^{3} + 1 |}{x^{2}} + C$

$\frac{1}{2} \log_{e} \frac{{(x^{3} + 1)}^{2}}{| x^{3} |} + C$

$\log_{e} | \frac{x^{3} + 1}{x} | + C$

$\frac{1}{2} \log_{e} \frac{| x^{3} + 1 |}{x^{2}} + C$

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

$\int \frac{2 x - \frac{1}{x^{2}}}{x^{2} + \frac{1}{x}} dx$

$x^{2} + \frac{1}{x} = t$

$(2 x - \frac{1}{x^{2}}) dx = dt$

$\int \frac{dt}{t} = ℓn (t) + C$

$= ℓn (x^{2} + \frac{1}{x}) + C$