To solve for the indefinite integrals, we'll have to integrate each given expression.
The first integral is ∫ 6x−76 dx.
The integral of 6x is 3*x^2, and the integral of -76 is -76x. You have to remember that when taking an indefinite integral, the result must include the constant of integration, which we've denoted as C. So, the answer yields: C + 3x^2 - 76x
The second integral to solve is ∫ (9−x)/(4x^3) dx. To simplify the process, the numerator and the denominator are divided separately: 9/(4x^3) - x/(4x^3). Following the rules of integration, the integral of 9/(4x^3) is -(9 - 2x)/(8x^2) and the result of x/(4x^3) is integrated as well. Similar to the previous integral, a constant of integration, C, is added. Thus, the full expression becomes: C - (9 - 2x)/(8x^2)
Lastly, for the integral ∫ (6 + e^x) * e^x dx, the two parts of the function are distributed: 6*e^x + e^(2x). The integral of 6*e^x is 6*e^x and the integral of e^(2x) is e^(2x)/2. The final solution, upon adding the constant of integration, gives us: C + e^(2x)/2 + 6*e^x
So, following the steps of basic integration rules - such as power rule and log rule - we can integrate polynomial functions, functions with rational exponents, and also transcendental functions which involve exponential terms. And always remember to add a constant of integration, denoted as C, to account for any constants that might have disappeared during the differentiation process which led to the original function.