Let `I=int_(0)^(1)(sinx)/(sqrtx) dx and f= int_(0)^(1)( cos x)/(sqrtx) dx`
Then , which one of the following is true ?

Options

A. `I gt (2)/(3) and f gt2`
B. `I lt (2)/(3) and f lt 2`
C. `I lt (2)/(3) and f gt 2`
D. `I gt (2)/(3) and f lt 2`

Answer

To solve the problem, we need to evaluate the integrals \( I \) and \( f \) defined as follows:

\[ I = \int_{0}^{1} \frac{\sin x}{\sqrt{x}} \, dx \]
\[ f = \int_{0}^{1} \frac{\cos x}{\sqrt{x}} \, dx \]

We will analyze both integrals and compare their values.

### Step 1: Analyze \( I \)

We know that for \( x \) in the interval \( [0, 1] \), \( \sin x \) is less than \( x \). Therefore, we can write:

\[ \sin x < x \quad \text{for } x \in [0, 1] \]

Dividing both sides by \( \sqrt{x} \) (which is positive for \( x \in (0, 1] \)), we have:

\[ \frac{\sin x}{\sqrt{x}} < \frac{x}{\sqrt{x}} = \sqrt{x} \]

Now, we can integrate both sides over the interval from 0 to 1:

\[ \int_{0}^{1} \frac{\sin x}{\sqrt{x}} \, dx < \int_{0}^{1} \sqrt{x} \, dx \]

Calculating the right-hand side:

\[ \int_{0}^{1} \sqrt{x} \, dx = \int_{0}^{1} x^{1/2} \, dx = \left[ \frac{x^{3/2}}{3/2} \right]_{0}^{1} = \frac{2}{3} \left( 1^{3/2} - 0^{3/2} \right) = \frac{2}{3} \]

Thus, we have:

\[ I < \frac{2}{3} \]

### Step 2: Analyze \( f \)

Next, we analyze \( f \):

We know that \( \cos x \) is always less than or equal to 1 for \( x \in [0, 1] \):

\[ \cos x \leq 1 \quad \text{for } x \in [0, 1] \]

Dividing both sides by \( \sqrt{x} \):

\[ \frac{\cos x}{\sqrt{x}} \leq \frac{1}{\sqrt{x}} = x^{-1/2} \]

Integrating both sides from 0 to 1:

\[ \int_{0}^{1} \frac{\cos x}{\sqrt{x}} \, dx \leq \int_{0}^{1} x^{-1/2} \, dx \]

Calculating the right-hand side:

\[ \int_{0}^{1} x^{-1/2} \, dx = \left[ \frac{x^{1/2}}{1/2} \right]_{0}^{1} = 2 \left( 1^{1/2} - 0^{1/2} \right) = 2 \]

Thus, we have:

\[ f < 2 \]

### Step 3: Conclusion

From our analysis, we have:

\[ I < \frac{2}{3} \quad \text{and} \quad f < 2 \]

Since \( \frac{2}{3} < 2 \), we can conclude that:

\[ I < f \]

### Final Result

Thus, the correct relationship is:

\[ I < f \]