Final answer:
The question asks for True or False evaluations of possible terms in the partial fractions decomposition of an integral. Terms must correspond to factors in the original denominator. Some terms are incorrectly notated and do not fit the proper form of partial fractions decomposition.
Step-by-step explanation:
The question revolves around calculating the partial fractions decomposition of a complex integral. First, we note that the given integral is a rational function where the numerator's degree is higher than the denominator's degree. Before partial fraction decomposition, we would typically perform polynomial division if necessary to simplify the integrand into a form suitable for partial fraction decomposition.
Assessing the choices given:
- False - The term A₁x+B₂/(x⁶+256)⁴ would not occur because the denominator's factors should match those in the original denominator of the integral.
- False - The term A₁ x² is not a fraction and does not represent a term in the decomposition of a rational function.
- True - The term Aix+Br/(x+4)² could occur as a part of the decomposition if (x-4)² is a factor in the denominator of the original integral.
- False - The term B/x+4 is incorrect notation and misrepresents the proper form of a partial fraction which would be B/(x+4) if x+4 were a factor of the denominator.
- True - The term B₁/x⁴ could potentially occur if x⁴ were a factor in the denominator of the original integral without multiplicity or as part of the decomposition of a higher power of x in the denominator.
Each term in a partial fractions decomposition corresponds to a factor in the denominator of the original integrand. The powers in the decomposition must also match the powers of the factors in the original denominator. Therefore, the correct terms in the complete partial fractions decomposition should directly relate to the factors and respective powers of the denominator of the original integrand.