Solve it: ∫ 1/x^1/3 dx?

Given:

The integral to be solved: ∫ (1 / x^1/3) dx

Concept Used:

The integral of xⁿ with respect to x is given by:

∫ xⁿ dx = (xⁿ⁺¹ / (n+1)) + C, where n ≠ -1.

In this problem, 1 / x^1/3 can be rewritten as x^-1/3.

Calculation:

Step 1: Rewrite the integral:

∫ (1 / x^1/3) dx = ∫ x^-1/3 dx

Step 2: Apply the formula ∫ xⁿ dx = (xⁿ⁺¹ / (n+1)) + C:

Here, n = -1/3, so n + 1 = 2/3.

⇒ ∫ x^-1/3 dx = (x^2/3 / (2/3)) + C

Step 3: Simplify the fraction:

x^2/3 / (2/3) = (3/2)x^2/3.

Step 4: Final Answer:

∫ (1 / x^1/3) dx = (3/2)x^2/3 + C

Conclusion:

∴ The correct answer is Option 1: (3/2)x^2/3 + C.