- Topic
- Antiderivatives
- Difficulty
- Capstone
- Suggested time
- 8 min
- Marks
- 5
- Theory
- CLP-1 §4.1Key results
Antiderivatives practice question 10
A function is defined for all and satisfies for every , with and . Find .
The domain comes in two separate pieces. Does one constant of integration have to serve both?
On each side of 0 separately, , and nothing forces the constant on the left to equal the constant on the right.
Use for the left constant and for the right one.
A whole number.
Theory CLP-1 §4.1Key results
Worked solution
Strategy
Two antiderivatives differ by a constant on an interval. The domain here is two intervals, so there are two independent constants, one per side, each fixed by its own condition.
Working
The general antiderivative of 1/x on each piece of the domain.
One condition per constant.
Evaluate on each side.
Sum.
Answer
The answer is 5.
Check
From x = 1 to e, and from −1 to −e, log|x| rises by 1, so f(−e) + f(e) = f(−1) + f(1) + 2 = 0 + 3 + 2 = 5.
Where this goes wrong
Writing f(x) = log|x| + C with a single C, which cannot satisfy both f(−1) = 0 and f(1) = 3. The “differ by a constant” theorem needs a single interval; across the gap at 0 the constants are independent.
Try a similar question
A function is defined for and satisfies there. Which of the following gives every such ?Saved in this browser only.