Integration of 1/(ax + b)
- is the one function the power rule can't handle (raising the power would make you divide by zero).
- So we use a different derivative instead: .
- Reverse that, then put in the same as before.
. 1 over a linear integrates to a log. The moment you see , write down a log.
The rule, and reading off a
- The bars let the inside be negative. You can't take of a negative number, and the bars fix that.
- In a definite integral where the inside stays positive, you can drop the bars.
Why this one gives a log, not a power
The power rule works for every power except . There , so you'd be dividing by zero. That one gap is exactly . A different function fills it: the logarithm. So when you integrate 1 over a linear, you can't use the power rule. The answer is a .
Worked examples
Worked example
Find
- It's a reciprocal of a linear, so the answer is a log. Read .
- Write and scale by the constant over : .
Answer
Worked example
Find
Check the sign: the number in front of is , so the is negative too:
Answer
Worked example
Evaluate (modelled on a P3 log integral)
- Antiderivative: so (the inside stays positive on , so no bars needed).
- Limits, top minus bottom: .
- Combine the logs with : .
Answer
Where the marks go
Common mistake
Don't drop the : , not . And read with its sign: has .
Common mistake
The log rule is only for a linear on the bottom. Something like is not a log. A squared bottom needs the methods of the next chapter, not this one.
Now you try
Find . (9709-style)
Your turn— tap to reveal the worked answer (9709-style)
1 over a linear, so a log: gives the antiderivative , then and :
Answer