Convergence of $\int_{-1}^{1}\frac{x-1}{x^{5/3}}dx$

Question

Find whether the following integral converges or diverges.$$\int_{-1}^{1}\frac{x-1}{x^{5/3}}dx$$

Attempt- I tried breaking the integral into two parts- one from -1 to 0 and other from 0 to 1 and then taking the limits but am facing problem while calculating the first limit i.e. from -1 to -$\epsilon.$ Kindly advise.

Answer 1

Hint: convince yourself that it is enough to investigate the convergence of $\int_{-1}^1 \frac{dx}{x^{5/3}}$. Then observe that the integrand is odd...

Answer 2

HINT:

$$\int\frac{x-1}{x^{\frac{5}{3}}}\space\text{d}x=$$

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Substitute $u=\sqrt[3]{x}$ and $\text{d}u=\frac{1}{3x^{\frac{2}{3}}}\space\text{d}x$:

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$$3\int\frac{u^3-1}{u^3}\space\text{d}u=$$ $$3\int\left(1-\frac{1}{u^3}\right)\space\text{d}u=$$ $$-3\int\frac{1}{u^3}\space\text{d}u+3\int 1\space\text{d}u=$$ $$\frac{3}{2u^2}+3u+\text{C}=$$ $$\frac{3}{2\left(\sqrt[3]{x}\right)^2}+3\sqrt[3]{x}+\text{C}=\frac{3(2x+1)}{2x^{\frac{2}{3}}}+\text{C}$$