Find the Antiderivative f(t)=10t^3+21

Problem

(∫_^)(10*t3+21*d(t))

Solution

1. Identify the integral of the sum as the sum of the integrals using the linearity property.

(∫_^)((10*t3+21)*d(t))=(∫_^)(10*t3*d(t))+(∫_^)(21*d(t))

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

(∫_^)(10*t3*d(t))=10⋅(t(3+1))/(3+1)

3. Simplify the expression for the first term.

10⋅(t4)/4=(5*t4)/2

4. Integrate the constant term using the rule (∫_^)(k*d(t))=k*t

(∫_^)(21*d(t))=21*t

5. Combine the results and add the constant of integration C

(5*t4)/2+21*t+C

Final Answer

(∫_^)(10*t3+21*d(t))=(5*t4)/2+21*t+C

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