This question involves finding an antiderivative through fundamental integration principles. To find an antiderivative of a(t) = 4t³ - 9t, we integrate each term using the power rule: ∫tⁿ dt = t^(n+1)/(n+1) + C. Integrating 4t³ gives 4·t⁴/4 = t⁴, and integrating -9t gives -9·t²/2 = -(9/2)t². Therefore, the antiderivative is t⁴ - (9/2)t² + C. Choice E (t⁴ - 9t² + C) incorrectly integrates the second term, forgetting to divide by the new power—a common algebraic error. The key is to systematically apply the power rule: add 1 to the exponent and divide by that new exponent.

Finding an antiderivative requires reversing the differentiation process using integration rules. For a(t) = 8t³ + 5, we apply the power rule: the antiderivative of 8t³ is 8t⁴/4 = 2t⁴, and the antiderivative of 5 is 5t. Therefore, the complete antiderivative is 2t⁴ + 5t + C, where C is the constant of integration. Choice B (2t⁴ + 5t) is incorrect because it omits the constant C, which is essential for representing all possible antiderivatives that differ by a constant. When integrating, remember to add 1 to the exponent and divide by the new exponent, and always include the constant C unless working with definite integrals.

This problem requires finding an antiderivative by applying the power rule for integration to each term. For v(x) = 7/2·x⁵ - 3x², we integrate term by term: the antiderivative of 7/2·x⁵ is (7/2)·x⁶/6 = 7x⁶/12, and the antiderivative of -3x² is -3x³/3 = -x³. The complete antiderivative is 7/12·x⁶ - x³ + C. Choice D (7/12·x⁶ - x³) is tempting but incorrect because it lacks the constant of integration C, which represents the infinite family of functions that all have the same derivative. The strategy for finding antiderivatives is to reverse differentiation: increase the power by 1, divide by the new power, and always include the constant C.

Finding the antiderivative involves basic reasoning about reversing the power rule for differentiation. To find the antiderivative of $18t^2$, increase the exponent to 3 and divide by 3, yielding $18(1/3)t^3$ = $6t^3$. For $5t^4$, increase to 5 and divide by 5, resulting in $t^5$. Combine these terms and add the constant C. A tempting distractor is choice C, which multiplies by the original coefficients without proper division. To master antiderivatives, always increase the exponent by one, divide by the new exponent, and include the constant C for indefinite integrals.

This question tests the skill of finding antiderivatives by reversing the power rule for differentiation. To find the antiderivative of p(t) = 7 - 9t², increase the exponent of each term by one and divide by the new exponent, resulting in 7t - (9/3)t³ + C. This process reverses differentiation because differentiating 7t - 3t³ + C yields 7 - 9t², matching the original population change rate. The constant C accounts for the fact that differentiation eliminates constants, so antiderivatives differ by a constant. A tempting distractor like choice B fails because it incorrectly uses a coefficient of -9 instead of -3 for the t³ term. Always remember that integration is the reverse of differentiation, so check your antiderivative by differentiating it back to see if you get the original function.

Finding the antiderivative involves basic reasoning about reversing the power rule for differentiation. To find the antiderivative of 1, increase the exponent from 0 to 1 and divide by 1, yielding t. For $-4t^7$, increase to 8 and divide by 8, resulting in $-(4/8)t^8$ = $-(1/2)t^8$. For $2t^3$, increase to 4 and divide by 4, giving $(1/2)t^4$, and add C. A tempting distractor is choice C, which omits division by the new exponents. To master antiderivatives, always increase the exponent by one, divide by the new exponent, and include the constant C for indefinite integrals.

Finding the antiderivative involves basic reasoning about reversing the power rule for differentiation. To find the antiderivative of $-2t^{-3}$, increase the exponent to -2 and divide by -2, yielding $-2(-1/2)t^{-2}$ = $t^{-2}$. There are no other terms, so add the constant C. This reverses the differentiation process for negative exponents. A tempting distractor is choice B, which incorrectly applies a negative sign. To master antiderivatives, always increase the exponent by one, divide by the new exponent, and include the constant C for indefinite integrals.

Finding the antiderivative involves basic reasoning about reversing the power rule for differentiation. To find the antiderivative of $3x^{-2}$, increase the exponent to -1 and divide by -1, yielding $3(-1)x^{-1}$ = $-3x^{-1}$. There are no other terms, so add the constant C. This reverses the differentiation of -3/x. A tempting distractor is choice B, which gets the sign wrong. To master antiderivatives, always increase the exponent by one, divide by the new exponent, and include the constant C for indefinite integrals.

This question tests the skill of finding antiderivatives by reversing the power rule for differentiation. To find the antiderivative of P(t) = 9t⁴ - 12t³, increase the exponent of each term by one and divide by the new exponent, resulting in (9/5)t⁵ - (12/4)t⁴ + C. This process reverses differentiation because differentiating (9/5)t⁵ - 3t⁴ + C yields 9t⁴ - 12t³, matching the original power function. The constant C accounts for the fact that differentiation eliminates constants, so antiderivatives differ by a constant. A tempting distractor like choice C fails because it incorrectly uses coefficients of 9 and -12 without dividing properly. Always remember that integration is the reverse of differentiation, so check your antiderivative by differentiating it back to see if you get the original function.

Finding the antiderivative involves basic reasoning about reversing the power rule for differentiation. To find the antiderivative of $5t^{2/3}$, increase the exponent to $5/3$ and divide by $5/3$, yielding $5(3/5)t^{5/3} = 3t^{5/3}$. There are no other terms, so add the constant C. This handles fractional exponents correctly. A tempting distractor is choice B, which forgets to divide properly. To master antiderivatives, always increase the exponent by one, divide by the new exponent, and include the constant C for indefinite integrals.