Question

The integral \( \int \frac{x+1}{x^{1/2}} \, dx \) is equal to:

• A. \( -x^{3/2} + x^{1/2} + c \)
• B. \( x^{1/2} \)
• C. \( \frac{2}{3}x^{3/2} + 2x^{1/2} + c \)
• D. \( x^{3/2} + x^{1/2} + c \)
• E. \( x^{3/2} \)

Hint:

Always simplify fractions before integrating. Converting roots to fractional exponents (\( \sqrt{x} = x^{1/2} \)) and moving variables to the numerator (\( 1/x^n = x^{-n} \)) makes the power rule easy to use.

Answer 1

The Correct Option is C

Concept: To integrate a rational expression where the denominator is a single term (monomial), it is often easiest to simplify the integrand by dividing each term in the numerator by the denominator. Once simplified into a sum of power functions, we apply the power rule of integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + c \).

Step 1: Simplify the integrand using laws of exponents.
Divide both terms in the numerator by \( x^{1/2} \): \[ \frac{x+1}{x^{1/2}} = \frac{x}{x^{1/2}} + \frac{1}{x^{1/2}} \] \[ = x^{1 - 1/2} + x^{-1/2} \] \[ = x^{1/2} + x^{-1/2} \]

Step 2: Apply the power rule to each term.
Integrate the simplified terms: \[ \int (x^{1/2} + x^{-1/2}) \, dx = \int x^{1/2} \, dx + \int x^{-1/2} \, dx \] Applying \( \frac{x^{n+1}}{n+1} \): \[ = \frac{x^{1/2+1}}{1/2+1} + \frac{x^{-1/2+1}}{-1/2+1} + c \] \[ = \frac{x^{3/2}}{3/2} + \frac{x^{1/2}}{1/2} + c \] \[ = \frac{2}{3}x^{3/2} + 2x^{1/2} + c \]