Evaluate the Integral

Problem

(∫_0^3)((3−t)√(,t)*d(t))

Solution

1. Rewrite the integrand by expressing the square root as a fractional power.

√(,t)=t(1/2)

2. Distribute the term t(1/2) into the parentheses.

(3−t)*t(1/2)=3*t(1/2)−t(3/2)

3. Apply the power rule for integration, which states (∫_^)(tn*d(t))=(t(n+1))/(n+1) to each term.

(∫_^)((3*t(1/2)−t(3/2))*d(t))=3(t(3/2))/(3/2)−(t(5/2))/(5/2)

4. Simplify the coefficients of the resulting expression.

2*t(3/2)−2/5*t(5/2)

5. Evaluate the definite integral at the upper limit t=3 and the lower limit t=0

(2*(3)(3/2)−2/5*(3)(5/2))−(2*(0)(3/2)−2/5*(0)(5/2))

6. Simplify the numerical expression using the property 3(5/2)=3⋅3(1/2)=9√(,3) and 3(3/2)=3√(,3)

2*(3√(,3))−2/5*(9√(,3))

7. Combine the terms by finding a common denominator.

6√(,3)−(18√(,3))/5

(30√(,3))/5−(18√(,3))/5=(12√(,3))/5

Final Answer

(∫_0^3)((3−t)√(,t)*d(t))=(12√(,3))/5

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