Evaluate the Integral

Problem

(∫_0^5)(√(,y+4)*d(y))

Solution

1. Identify the integral as a definite integral of a radical function.

2. Rewrite the integrand using a fractional exponent to make it easier to apply the power rule.

(∫_0^5)((y+4)(1/2)*d(y))

3. Apply the power rule for integration, which states that (∫_^)(un*d(u))=(u(n+1))/(n+1) Here, u=y+4 and d(u)=d(y)

[((y+4)(3/2))/(3/2)]50

4. Simplify the constant coefficient by multiplying by the reciprocal.

[2/3*(y+4)(3/2)]50

5. Substitute the upper limit of integration (y=5 and the lower limit of integration (y=0 into the expression.

2/3*(5+4)(3/2)−2/3*(0+4)(3/2)

6. Evaluate the numerical terms by calculating the square roots and then cubing them.

2/3*(9)(3/2)−2/3*(4)(3/2)

7. Simplify the powers: 9(3/2)=(√(,9))3=3=27 and 4(3/2)=(√(,4))3=2=8

2/3*(27)−2/3*(8)

8. Calculate the final values and subtract.

18−16/3

9. Find a common denominator to reach the final result.

54/3−16/3=38/3

Final Answer

(∫_0^5)(√(,y+4)*d(y))=38/3

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