what is FTC?
use ftc to evaluate a definite integral with basic integrand.
f looking sign, integral bottom number 0 top number 2 (4x^3+1) dx=
FTC stands for the Fundamental Theorem of Calculus. It is a fundamental concept in calculus that allows us to evaluate definite integrals by finding antiderivatives.
To evaluate the definite integral using the FTC, follow these steps:
1. Identify the function and the bounds: In this case, the function is (4x^3 + 1) and the integral is from 0 to 2.
2. Find the antiderivative: To do this, use the power rule for integration. For each term, add 1 to the exponent and divide by the new exponent. The antiderivative of 4x^3 is x^4 (divide the exponent 4 by the new exponent 4). The antiderivative of 1 is x (divide the exponent 1 by the new exponent 1).
3. Evaluate the antiderivative at the upper bound: Substitute the upper bound (2) into the antiderivative function. Evaluate x^4 at 2 to get 2^4, which is 16. Evaluate x at 2 to get 2.
4. Evaluate the antiderivative at the lower bound: Substitute the lower bound (0) into the antiderivative function. Evaluate x^4 at 0 to get 0. Evaluate x at 0 to get 0.
5. Subtract the result of step 4 from the result of step 3: Subtract 0 from 16 to get 16.
Therefore, the value of the definite integral ∫[0,2] (4x^3 + 1) dx is equal to 16.