Question

Find the most general antiderivative of the function g(t) = 8t . t^2 .t . Check your answer by differentiation. Use C for the constant of the antiderivative.

A) 2t^4+ C
B) 8/4t^4 + C
C) 2t^5 + C
D) 8/3t^5 + C

Answer 1

The most general antiderivative of g(t) = 8t * t^2 * t is 2t^5 + C. To check the answer, we can differentiate it and confirm that the derivative matches the original function.

Step-by-step explanation:

To find the most general antiderivative of the function g(t) = 8t * t^2 * t, we need to apply the power rule for integration. The power rule states that for a term of the form x^n, the antiderivative is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration. In this case, we have g(t) = 8t * t^2 * t, which can be simplified to 8t^4. Applying the power rule, we get the antiderivative of g(t) as (1/(4+1)) * 8t^(4+1) + C, which simplifies to 2t^5 + C. Therefore, the correct answer is option C) 2t^5 + C.

To check our answer, we can differentiate 2t^5 + C with respect to t. The derivative of 2t^5 is 10t^4. Any constant C differentiates to 0, since it does not involve t. Therefore, the derivative of 2t^5 + C is 10t^4, which matches the original function g(t) = 8t * t^2 * t. So our answer is confirmed to be correct.