Final Answer:
The indefinite integral of 13x² - 7 dx is 13/3 x³ - 7x + C, where C represents the constant of integration.
Step-by-step explanation:
To find the indefinite integral ∫(13x² - 7) dx, integrate term by term using the power rule of integration. The integral of 13x² with respect to x is (13/3) x³, obtained by adding 1 to the exponent (2 + 1 = 3) and dividing the coefficient by the new exponent (13/3). For the constant term -7, integrating with respect to x gives -7x.
The result of integrating both terms is (13/3) x³ - 7x. Always remember to add the constant of integration, represented by 'C,' as it accounts for any constant that might have been present in the original function but does not appear explicitly in the derivative. Therefore, the final solution for the indefinite integral of 13x² - 7 dx is 13/3 x³ - 7x + C, where C represents the constant term.
This integral represents a family of functions that, when differentiated, would yield 13x² - 7. The constant of integration allows for the inclusion of any constant value that may have been present in the original function but was not explicitly stated. Integrating using the power rule is fundamental in finding antiderivatives or indefinite integrals of polynomial functions, applying the reverse operation of differentiation to find the original function from its derivative.