∫ dx/ x ^a =

The correct option is:
$\frac{3}{4}{x}^{\frac{4}{3}}+C\text{for}a=-\frac{1}{3}$

Step-by-Step Explanation:

We want to evaluate the general integral:
$\int \frac{dx}{x^a}$

Using exponent properties, we can rewrite the integrand by bringing the term in the denominator to the numerator with a negative exponent:
$\int x^{-a}dx$

For the specific value of $a=-\frac{1}{3}$, the exponent becomes:
$-a=-(-\frac{1}{3})=\frac{1}{3}$

Substituting this value back into the integral gives:
$\int x^{\frac{1}{3}}dx$

Next, we apply the power rule for integration, which states that for any constant $n\ne -1$:
$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$
where C is the constant of integration.

Setting $n=\frac{1}{3}$, we calculate the new power term:
$n+1=\frac{1}{3}+1=\frac{4}{3}$

Applying this to the power rule:
$\int x^{\frac{1}{3}}dx=\frac{x^{\frac{4}{3}}}{\frac{4}{3}}+C$

Simplifying the coefficient in front by multiplying by the reciprocal of the denominator:
$\frac{3}{4}{x}^{\frac{4}{3}}+C$

This successfully derives the given correct option.