\(\int_{4}^{1} x \sqrt x dx \)  = ?

Given:

We need to evaluate the integral: \(\int_{4}^{1} x \sqrt x dx \) 

Concept Used:

The integral of a function can be solved using the power rule:

If f(x) = xⁿ, then ∫xⁿ dx = (xⁿ⁺¹) / (n+1) + C, where n ≠ -1.

Here, x√x = x^3/2, so we integrate x^3/2.

Calculation:

Step 1: Write the integral in terms of powers of x.

\(\int_{4}^{1} x \sqrt x dx \)  = \(\int_{4}^{1} x^{\frac {3}{2}} dx \) 

Step 2: Apply the power rule of integration.

⇒ ∫x^3/2 dx = (x^{(3/2) + 1}) / ((3/2) + 1)

⇒ ∫x^3/2 dx = (x^5/2) / (5/2) ⇒ ∫x^3/2 dx = (2/5) x^5/2

Step 3: Apply the limits of the definite integral (from 4 to 1).

⇒ \(\int_{4}^{1} x^{\frac {3}{2}} dx \) = [(2/5) x^5/2] ₄¹

Step 4: Substitute the limits into the equation.

For x = 4: (2/5) × 4^5/2 = (2/5) × (4^1/2)⁵ = (2/5) × (2)⁵ = (2/5) × 32 = 64/5 = 12.8

For x = 1: (2/5) × 1^5/2 = (2/5) × 1 = 2/5 = 0.4

Step 5: Subtract the results.

⇒ \(\int_{4}^{1} x^{\frac {3}{2}} dx \)  = 12.8 - 0.4 = 12.4

Conclusion:

∴ The value of the integral is 12.4.

However, according to the provided options, the correct answer given is Option 1: 7.