Question

All values of \(a\) for which the inequality
\[ \frac{1}{\sqrt{a}} \int_{1}^{a} \left( \frac{3}{2} \sqrt{x} + 1 - \frac{1}{\sqrt{x}} \right) dx < 4 \]
is satisfied, lie in the interval.

• A. (1,2)
• B. (0,3)
• C. (0,4)
• D. (1,4)

Hint:

To solve inequalities involving integrals, break the integral into simpler terms and simplify the resulting expressions for easier evaluation.

Answer 1

The Correct Option is C

Step 1: Separate the integral into terms:

\[ I(a) = \int_{1}^{a} \left( \frac{3}{2} \sqrt{x} + 1 - \frac{1}{\sqrt{x}} \right) dx \]

Step 2: Decompose into individual integrals:

\[ I(a) = \int_{1}^{a} \frac{3}{2} \sqrt{x} \, dx + \int_{1}^{a} 1 \, dx - \int_{1}^{a} \frac{1}{\sqrt{x}} \, dx \]

Step 3: Calculate each integral:

\[ \int_{1}^{a} \frac{3}{2} \sqrt{x} \, dx = \left[ \frac{3}{2} \cdot \frac{2}{3} x^{3/2} \right]_{1}^{a} = a^{3/2} - 1 \] \[ \int_{1}^{a} 1 \, dx = a - 1 \] \[ \int_{1}^{a} \frac{1}{\sqrt{x}} \, dx = \left[ 2\sqrt{x} \right]_{1}^{a} = 2\sqrt{a} - 2 \]

Step 4: Substitute the results and simplify the inequality:

\[ \frac{1}{\sqrt{a}} \left( a^{3/2} - 1 + a - 1 - 2\sqrt{a} + 2 \right) < 4 \]

Step 5: Solve the inequality, which leads to \( a \in (0, 4) \).