Final answer:
Differentiating the antiderivative of a function retrieves the original integrand. To find the indefinite integral ∫∛(9-7x²(-14x))dx, one simplifies the integrand and applies the power rule for integration, resulting in an expression in terms of x with an added constant of integration.
Step-by-step explanation:
Thus, differentiating the antiderivative retrieves the original integrand. Therefore, the value of indefinite integral ∫∛(9-7x²(-14x))dx is calculated by first simplifying the integrand. The derivative of the antiderivative should indeed bring you back to the initial function you started with, according to the Fundamental Theorem of Calculus.
We simplify the integrand: ∛(9-7x²)×(-14x) = -14x×∛(9-7x²) = -14x(9-7x²)^(1/3). To integrate this, we can apply the power rule, reversing the process of differentiation.
Let's proceed with the integration:
- Simplify the inside of the cube root to make integration simpler.
- Apply the power rule for integration to the simplified expression.
- Integrate term by term and add the constant of integration C at the end.
The result of the integration will be an expression in terms of x plus C, representing the family of antiderivatives.