Final answer:
To find the integral of x⁹sin(x⁵) as a power series, we can use the power series expansion for the sine function and substitute x⁵ for x. Multiplying the resulting series by x⁹ gives us the power series for x⁹sin(x⁵).
Step-by-step explanation:
To find the integral of x⁹sin(x⁵) as a power series, we can use the power series expansion for the sine function. We know that the power series for sin(x) is given by:
sin(x) = x - (x³/3!) + (x⁵/5!) - (x⁷/7!) + ...
By substituting x⁵ for x in this series, we have:
sin(x⁵) = x⁵ - (x¹⁵/3!) + (x²⁵/5!) - (x³⁵/7!) + ...
Now, we can multiply the power series for sin(x⁵) by x⁹ to obtain the power series for x⁹sin(x⁵):
x⁹sin(x⁵) = x¹⁴ - (x²⁴/3!) + (x³⁴/5!) - (x⁴⁴/7!) + ...