Final answer:
To find the general antiderivative of the function f(x) = 6x⁹ - 4x⁶ + 15x³ - 6csc²(x), we need to apply the power rule to each polynomial term and recognize the antiderivative of the trigonometric term. The antiderivative is F(x) = 0.6x¹⁰ - 0.57x⁷ + 3.75x⁴ + 6cot(x) + C, with C as the constant of integration.
Step-by-step explanation:
The student has asked to find the most general antiderivative of the function f(x) = 6x⁹ - 4x⁶ + 15x³ - 6csc²(x). For the polynomial part of f(x), we apply the power rule for antiderivatives, which states if f(x) = x⁹, then F(x) = \frac{1}{n+1}x^{n+1} + C, where C is the constant of integration. For the trigonometric part -6csc²(x), the antiderivative is 6cot(x), because d/dx [cot(x)] = -csc²(x).
Now we will find the antiderivative of each individual term:
- For 6x⁹, the antiderivative is \(rac{6}{10}x^{10}\) or 0.6x¹⁰.
- For -4x⁶, the antiderivative is \(rac{-4}{7}x^{7}\) or -0.57x⁷.
- For 15x³, the antiderivative is \(rac{15}{4}x^{4}\) or 3.75x⁴.
- For -6csc²(x), the antiderivative is 6cot(x).
Combining all these results, the most general antiderivative of f(x) is:
F(x) = 0.6x¹⁰ - 0.57x⁷ + 3.75x⁴ + 6cot(x) + C
where C is the constant of integration.