\(\int_{1}^{4} x \sqrt{x} dx \)

Concept:

Integration of Algebraic Functions:

• The given integral is a definite integral involving powers of x.
• We simplify the expression using the identity: √x = x^1/2.
• Use the rule: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), where n ≠ −1.
• For definite integrals, apply the limits after integration: \( \int_a^b f(x) dx = F(b) - F(a) \)

Calculation:

Given,

\( \int_{1}^{4} x \sqrt{x} dx \)

⇒ x × √x = x × x^1/2 = x^3/2

⇒ \( \int_{1}^{4} x^{3/2} dx \)

⇒ \( \left[\frac{x^{5/2}}{5/2}\right]_{1}^{4} \)

⇒ \( \left[\frac{2}{5} x^{5/2}\right]_{1}^{4} \)

⇒ \( \frac{2}{5}(4^{5/2} - 1^{5/2}) \)

⇒ 4^5/2 = (√4)⁵ = 2⁵ = 32

⇒ 1^5/2 = 1

⇒ \( \frac{2}{5}(32 - 1) = \frac{2}{5} \times 31 = \frac{62}{5} \) = 12.4

∴ The value of the definite integral is 12.4