- Topic
- Antiderivatives
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
- Theory
- CLP-1 §4.1Key results
Antiderivatives practice question 2
Which of the following is the most general antiderivative of ?
Ask “what differentiates to ?” and “what differentiates to a constant?”. Then check your answer by differentiating it.
, so watch the sign in front of the cosine. “Most general” also means the family must allow every constant.
Theory CLP-1 §4.1Key results
Worked solution
Strategy
Read the differentiation rules backwards: cos differentiates to −sin, so −cos differentiates to sin; a constant is the derivative of a linear term. “Most general” means the + C is part of the answer.
Working
Each piece checks out.
Any two antiderivatives differ by a constant, so the family needs C.
Answer
The answer is negative 3 cosine x minus 2x plus a constant.
Check
Differentiating −3 cos x − 2x + C gives 3 sin x − 2 whatever C is, while 3 cos x − 2x + C gives −3 sin x − 2.
Where this goes wrong
Choosing 3cos x − 2x + C, with the sign of the cosine wrong. Differentiating that gives −3 sin x − 2, not 3 sin x − 2; always differentiate to check.
Leans on
The derivative of cos, from Exponential and trigonometric functions
Try a similar question
Find the function with and , and evaluate .Saved in this browser only.