求不定积分∫(x+7)/(x+8)^5 dx，请教简便求解方法

Hey there! I totally get how integrals like this can feel tricky at first, but there are two super straightforward methods to solve this without overcomplicating things. Let me break them down for you:

Method 1: U-Substitution (Classic Go-To) #

This method works perfectly when you have a linear term raised to a power in the denominator. Here's the step-by-step:

• Let u = x + 8. That means x = u - 8, and dx = du (since the derivative of x+8 is just 1).
• Rewrite the numerator using this substitution: x + 7 = (u - 8) + 7 = u - 1.
• Plug everything into the integral to simplify it into basic power functions:

  ∫(u - 1)/u^5 du = ∫(u/u^5 - 1/u^5) du = ∫(u^(-4) - u^(-5)) du
  

• Apply the power rule for integration (∫x^n dx = x^(n+1)/(n+1) + C for n ≠ -1):

  ∫u^(-4) du - ∫u^(-5) du = (-1/(3u³)) - (-1/(4u⁴)) + C = -1/(3u³) + 1/(4u⁴) + C
  

• Swap back u = x + 8 to get the final result in terms of x:

  -1/(3(x+8)³) + 1/(4(x+8)⁴) + C
  

Method 2: Rewrite the Numerator Directly (Even Faster!) #

You don't even need substitution if you spot how to rearrange the numerator to match the denominator:

• Notice that x + 7 = (x + 8) - 1. Use this to split the fraction:

  (x+7)/(x+8)^5 = (x+8 - 1)/(x+8)^5 = (x+8)/(x+8)^5 - 1/(x+8)^5 = 1/(x+8)^4 - 1/(x+8)^5
  

• Integrate term by term using the power rule (for linear terms like x+a, the integral of 1/(x+a)^n is -1/((n-1)(x+a)^(n-1)) + C):

  ∫1/(x+8)^4 dx - ∫1/(x+8)^5 dx = -1/(3(x+8)³) + 1/(4(x+8)^4) + C
  

Both methods land you the same result, and the second one is lightning-fast once you catch the numerator trick. The key here is avoiding overcomplicated methods like integration by parts—instead, lean into simplifying the integrand into basic functions you already know how to integrate.

内容的提问来源于stack exchange，提问作者Marina Ezzat