求∫sin(x°)dx的积分结果，验证猜想答案的正确性

Hey there! Let's work through this integral confusion together—your question about handling the degree symbol in ∫sin(x°)dx is totally valid, since calculus relies on radians for trigonometric operations, which is probably where the mix-up is happening.

First: Understand the Degree-to-Radian Conversion #

All standard derivative and integral formulas for trigonometric functions (like ∫sin(x)dx = -cos(x) + C) assume the input is in radians, not degrees. So we first need to convert x° to its equivalent radian value:

1 degree = π/180 radians, so x° = (πx)/180 radians.

Step-by-Step Integration #

Rewrite the original integral using radians:

∫sin(x°)dx = ∫sin( (πx)/180 ) dx

Now use substitution (the reverse of the chain rule) to solve this:

• Let u = (πx)/180. Taking the derivative with respect to x gives du/dx = π/180, so rearranged: dx = (180/π) du.
• Substitute into the integral:

  ∫sin(u) * (180/π) du
  

• Factor out the constant 180/π and integrate sin(u) (which we know gives -cos(u)):

  (180/π)*(-cos(u)) + C = -(180/π)cos(u) + C
  

• Substitute back u = (πx)/180 (or just write it as x° since they're equivalent):

  -(180/π)cos(x°) + C
  

Why Your Guessed Answer Isn't Correct #

Your proposed answer -cosx/x° doesn't work for two key reasons:

• The degree symbol ° is a unit, not a variable term—you can't divide by it like it's part of the function.
• You didn't account for the degree-to-radian conversion. Without converting to radians, you can't apply the standard integral formula for sin(x) directly to sin(x°).

内容的提问来源于stack exchange，提问作者Anshi