Evaluate the Integral

Problem

(∫_1^4)(√(,x)*d(x))

Solution

1. Rewrite the integrand using a fractional exponent to make it easier to apply the power rule for integration.

√(,x)=x(1/2)

2. Apply the power rule for integration, which states that (∫_^)(xn*d(x))=(x(n+1))/(n+1) for n≠−1

(∫_^)(x(1/2)*d(x))=(x(3/2))/(3/2)

3. Simplify the resulting expression by multiplying by the reciprocal of the denominator.

(x(3/2))/(3/2)=2/3*x(3/2)

4. Evaluate the definite integral at the upper limit of 4 and the lower limit of 1 using the Fundamental Theorem of Calculus.

[2/3*x(3/2)]41=2/3*(4)(3/2)−2/3*(1)(3/2)

5. Calculate the numerical values by noting that 4(3/2)=(√(,4))3=2=8 and 1(3/2)=1

2/3*(8)−2/3*(1)=16/3−2/3

6. Subtract the fractions to find the final result.

16/3−2/3=14/3

Final Answer

(∫_1^4)(√(,x)*d(x))=14/3

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