Evaluate the Integral

Problem

(∫_−1^1)(√(3,t−2)*d(t))

Solution

1. Rewrite the radical expression using a fractional exponent to make it easier to integrate.

(∫_−1^1)((t−2)(1/3)*d(t))

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

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

3. Simplify the exponent and the denominator.

((t−2)(4/3))/(4/3)=(3*(t−2)(4/3))/4

4. Evaluate the definite integral by plugging in the upper limit 1 and the lower limit −1

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

5. Calculate the values inside the parentheses.

(3*(−1)(4/3))/4−(3*(−3)(4/3))/4

6. Simplify the terms using the property x(4/3)=(x4)(1/3) Note that (−1)4=1 and (−3)4=81

(3*(1)(1/3))/4−(3*(81)(1/3))/4

7. Factor out the common terms and simplify the cube root of 81 as 3√(3,3)

3/4*(1−√(3,81))

3/4*(1−3√(3,3))

Final Answer

(∫_−1^1)(√(3,t−2)*d(t))=(3−9√(3,3))/4

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