Evaluate the Integral integral of (1+3t)t^2 with respect to t

Problem

(∫_^)((1+3*t)*t2*d(t))

Solution

1. Distribute the term t2 into the parentheses to rewrite the integrand as a polynomial.

(∫_^)((t2+3*t3)*d(t))

2. Apply the sum rule for integration, which allows the integral of a sum to be evaluated as the sum of the integrals.

(∫_^)(t2*d(t))+(∫_^)(3*t3*d(t))

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

(t3)/3+3⋅(t4)/4+C

4. Simplify the expression by combining the coefficients.

1/3*t3+3/4*t4+C

Final Answer

(∫_^)((1+3*t)*t2*d(t))=1/3*t3+3/4*t4+C

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