Để khai triển Maclaurin của $x\sin x - \cos x$ đến bậc 6, ta thực hiện như sau:

1. Maclaurin của $\sin x$: $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} + O(x^7)$

2. Maclaurin của $\cos x$: $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + O(x^7)$

3. Biểu thức $x\sin x$: $x\sin x = x \cdot \left(x - \frac{x^3}{3!} + \frac{x^5}{5!}\right) = x^2 - \frac{x^4}{6} + \frac{x^6}{120} + O(x^7)$

4. Biểu thức $x\sin x - \cos x$: $x\sin x - \cos x = \left(x^2 - \frac{x^4}{6} + \frac{x^6}{120}\right) - \left(1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720}\right)$

5. Rút gọn đến bậc 6: $x\sin x - \cos x = x^2 - \frac{x^4}{6} + \frac{x^6}{120} - 1 + \frac{x^2}{2} - \frac{x^4}{24} + \frac{x^6}{720}$

   Gom các bậc tương ứng:

   • Hệ số $x^2$: $x^2 + \frac{x^2}{2} = \frac{3x^2}{2}$
   • Hệ số $x^4$: $-\frac{x^4}{6} - \frac{x^4}{24} = -\frac{5x^4}{24}$
   • Hệ số $x^6$: $\frac{x^6}{120} + \frac{x^6}{720} = \frac{7x^6}{720} = \frac{x^6}{103}$.

   Do đó, khai triển Maclaurin của $x\sin x - \cos x$ đến bậc 6 là:

   $$$$

6. Recall the expansions: [ \sin x = x - \frac{x^3}{6} + \frac{x^5}{120} + O(x^7)$$\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + O(x^7)$$

7. Compute $x \sin x$: $x \sin x = x \cdot \left(x - \frac{x^3}{6} + \frac{x^5}{120}\right)$ $x \sin x = x^2 - \frac{x^4}{6} + \frac{x^6}{120} + O(x^7)$

8. Compute $x \sin x - \cos x$: $x \sin x - \cos x = \left(x^2 - \frac{x^4}{6} + \frac{x^6}{120}\right) - \left(1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720}\right)$

9. Combine terms:

   • Constant term: $-1$
   • $x^2$ term: $x^2 - \left(-\frac{x^2}{2}\right) = x^2 + \frac{x^2}{2} = \frac{3x^2}{2}$
   • $x^4$ term: $-\frac{x^4}{6} - \frac{x^4}{24} = -\frac{4x^4}{24} - \frac{x^4}{24} = -\frac{5x^4}{24}$
   • $x^6$ term: $\frac{x^6}{120} - \frac{x^6}{720} = \frac{6x^6}{720} - \frac{x^6}{720} = \frac{5x^6}{720} = \frac{x^6}{144}$

   So: $x \sin x - \cos x = -1 + \frac{3x^2}{2} - \frac{5x^4}{24} + \frac{x^6}{144} + O(x^7)$

Final Answer:

$x \sin x - \cos x = -1 + \frac{3x^2}{2} - \frac{5x^4}{24} + \frac{x^6}{144}$

Questions for Deeper Understanding:

1. Can you confirm the general formula for the Maclaurin series expansion?
2. How is the factorial function involved in the derivation of Maclaurin series?
3. What role does truncation play in approximation?
4. What is the difference between Maclaurin and Taylor series?
5. How does the error term $O(x^7)$ affect practical applications?

Tip:

When performing Maclaurin expansions, align terms carefully to avoid sign and coefficient errors during subtraction.