Final Answer:
1) ∫(1/5)(x + 3) dx = (1/10)x^2 + (3/5)x + C, where C is the constant of integration.
2) ∫(3x² + 4x³)/(x³ + x⁴) dx = ln|x³ + x⁴| + C, where C is the constant of integration.
3) ∫(3x² - 2x)/(1 - x² + x³) dx = ln|1 - x² + x³| + C, where C is the constant of integration.
4) ∫6z/(z² - 6) dz = 3ln|z² - 6| + C, where C is the constant of integration.
Step-by-step explanation:
1) For the first integral, ∫(1/5)(x + 3) dx, we use the power rule for integration. The antiderivative of x is (1/2)x^2, and the antiderivative of 3 is 3x. Applying these, we get (1/10)x^2 + (3/5)x + C, where C is the constant of integration.
2) In the second integral, ∫(3x² + 4x³)/(x³ + x⁴) dx, we simplify the expression inside the integral, factor out an x³ from the numerator, cancel out common factors, and integrate. The natural logarithm ln|x³ + x⁴| appears as the antiderivative, and we add the constant of integration, C.
3) For the third integral, ∫(3x² - 2x)/(1 - x² + x³) dx, we manipulate the expression by factoring the denominator and canceling out common factors. The integral simplifies to ln|1 - x² + x³| + C, and we include the constant of integration, C.
4) In the fourth integral, ∫6z/(z² - 6) dz, we perform partial fraction decomposition to simplify the integrand. The antiderivative involves the natural logarithm, and the final result is 3ln|z² - 6| + C, with the constant of integration, C.
These solutions follow standard techniques in calculus, ensuring accurate integration results.