how do you find the answer to (2t-1)^3 dt on the integral from x^2 to x^3
To find the answer to the integral of (2t-1)^3 dt from x^2 to x^3, you can follow these steps:
1. First, expand the expression (2t-1)^3. It can be written as (2t-1)(2t-1)(2t-1), which simplifies to 8t^3 - 12t^2 + 6t - 1.
2. Now, we have the integral of the expanded expression: ∫(8t^3 - 12t^2 + 6t - 1) dt.
3. To evaluate this definite integral, we need to find the antiderivative (or integral) of the polynomial function (8t^3 - 12t^2 + 6t - 1).
4. Applying the power rule, we can find the antiderivative of each term separately. The antiderivative of 8t^3 is (8/4) * t^4 = 2t^4, the antiderivative of -12t^2 is (-12/3) * t^3 = -4t^3, the antiderivative of 6t is (6/2) * t^2 = 3t^2, and the antiderivative of -1 is -t.
5. Now, we have the antiderivative of the polynomial: 2t^4 - 4t^3 + 3t^2 - t.
6. Next, we evaluate the antiderivative at both limits of integration, which are x^2 and x^3.
7. Substituting x^2 into the antiderivative expression, we get: 2(x^2)^4 - 4(x^2)^3 + 3(x^2)^2 - (x^2).
8. Similarly, substituting x^3 into the antiderivative expression, we get: 2(x^3)^4 - 4(x^3)^3 + 3(x^3)^2 - (x^3).
9. Subtracting the result obtained in step 7 from the result in step 8, we have the final answer: (2(x^3)^4 - 4(x^3)^3 + 3(x^3)^2 - (x^3)) - (2(x^2)^4 - 4(x^2)^3 + 3(x^2)^2 - (x^2)).
10. Simplify the expression obtained in step 9 to obtain the final answer.
Following these steps will help you find the answer to the given integral.