Question

If $\int \frac{5\sqrt{x}(4x^2 - 1)}{x} dx = 8\sqrt{x^5} + a\sqrt{x} + C$, where $a$ is a constant and $C$ is the constant of integration, then the value of $a$ is

• A. 20
• B. -20
• C. -10
• D. 10
• E. -8

Hint:

When the denominator is a single power of $x$, always split the fraction and use the power rule \( \int x^n dx = \frac{x^{n+1}}{n+1} \). It's simpler and less prone to error than substitution.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To solve this, distribute the terms in the numerator and simplify the powers of $x$. Then integrate term by term.

Step 2: Detailed Explanation:
Simplify the integrand:
\[ \frac{5\sqrt{x}(4x^2 - 1)}{x} = \frac{20x^{5/2} - 5x^{1/2}}{x} = 20x^{3/2} - 5x^{-1/2} \]
Now, integrate:
\[ I = \int (20x^{3/2} - 5x^{-1/2}) dx \]
\[ I = 20 \left( \frac{x^{5/2}}{5/2} \right) - 5 \left( \frac{x^{1/2}}{1/2} \right) + C \]
\[ I = 20 \left( \frac{2}{5} x^{5/2} \right) - 5 (2 x^{1/2}) + C \]
\[ I = 8x^{5/2} - 10x^{1/2} + C \]
Rewrite in the given radical form:
\[ I = 8\sqrt{x^5} - 10\sqrt{x} + C \]
Comparing this with \( 8\sqrt{x^5} + a\sqrt{x} + C \), we get:
\[ a = -10 \]

Step 3: Final Answer:
The value of $a$ is -10.