Question

Rewrite, Integrate, and Simplify the integral

∫ (1 / x√x) dx
a) 2/3x³/² + C
b) 1/2x³/² + C
c) 2/5x⁵/² + C
d) 1/3x³/² + C

Answer 1

The correct integral of 1 / x√x is 2/3x³/² + C, which corresponds to option a).

Step-by-step explanation:

The student asked to rewrite, integrate, and simplify the integral ∫ (1 / x√x) dx. To solve this integral, we rewrite the integrand in a simpler form. Recognizing that √x is x^(1/2), we can express the integral as ∫ x^(-3/2) dx. This is a standard power rule integration scenario, where the integral of x^n dx is (1/(n+1))x^(n+1) for n ≠ -1.

Applying the power rule with n = -3/2 gives us:

∫ x^(-3/2) dx = (1/(-3/2 + 1))x^(-3/2 + 1) + C

This simplifies to:

2/3x^(1/2) + C, which further simplifies to:

2/3x³/² + C, which corresponds to option a).