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:
We evaluate an indefinite integral algebraically. We simplify the integrand into power forms \(x^n\) and then apply the power rule of integration.
Step 2: Key Formula or Approach:
Distribute terms and use \(x^m / x^n = x^{m-n}\).
Integrate using \(\int x^n dx = \frac{x^{n+1}}{n+1}\).
Compare the resulting expression with the given format to find \(a\).
Step 3: Detailed Explanation:
Simplify the integrand:
\[ \frac{5x^{1/2}(4x^2 - 1)}{x} = \frac{20x^{5/2} - 5x^{1/2}}{x^1} \] \[ = 20x^{3/2} - 5x^{-1/2} \] Integrate each term:
\[ \int (20x^{3/2} - 5x^{-1/2}) dx = 20 \frac{x^{3/2 + 1}}{3/2 + 1} - 5 \frac{x^{-1/2 + 1}}{-1/2 + 1} + C \] \[ = 20 \frac{x^{5/2}}{5/2} - 5 \frac{x^{1/2}}{1/2} + C \] \[ = 20 \left(\frac{2}{5}\right) x^{5/2} - 5 \left(\frac{2}{1}\right) x^{1/2} + C \] \[ = 8x^{5/2} - 10x^{1/2} + C \] Convert fractional exponents back to radicals:
\[ = 8\sqrt{x^5} - 10\sqrt{x} + C \] Compare this with the given RHS: \(8 \sqrt{x^5} + a\sqrt{x} + C\).
We can clearly see that \(a = -10\).
Step 4: Final Answer:
The value of \(a\) is -10.