Evaluate the Integral

Problem

(∫_1^3)(3*t2+7*d(t))

Solution

1. Identify the integrand and the limits of integration. The integrand is 3*t2+7 and the interval is [1,3]

2. Find the antiderivative of the function using the power rule for integration, which states (∫_^)(tn*d(t))=(t(n+1))/(n+1)

3. Apply the power rule to each term. The antiderivative of 3*t2 is t3 and the antiderivative of 7 is 7*t

F(t)=t3+7*t

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit and subtracting the value at the lower limit.

[t3+7*t]31=(3+7*(3))−(1+7*(1))

5. Simplify the numerical expression.

(27+21)−(1+7)

48−8

40

Final Answer

(∫_1^3)(3*t2+7*d(t))=40

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