Evaluate the Integral

Problem

(∫_1^4)(1/√(,t)*d(t))

Solution

1. Rewrite the integrand using a power of t to make it easier to integrate.

1/√(,t)=t(−1/2)

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

(∫_^)(t(−1/2)*d(t))=(t(1/2))/(1/2)

3. Simplify the resulting expression before evaluating the limits.

(t(1/2))/(1/2)=2√(,t)

4. Apply the Fundamental Theorem of Calculus by evaluating the expression at the upper limit 4 and subtracting the evaluation at the lower limit 1

[2√(,t)]41=2√(,4)−2√(,1)

5. Calculate the final numerical values.

2*(2)−2*(1)=4−2

4−2=2

Final Answer

(∫_1^4)(1/√(,t)*d(t))=2

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