Science:Math Exam Resources/Courses/MATH110/April 2012/Question 09
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Question 09 

An antiderivative of a function ƒ is a function F satisfying for all x. Find two antiderivatives of the function

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Recall that is its own derivative. 
Hint 2 

Suppose you have one function that has a derivative of . Can you think of an easy function that has derivative zero so that when you add to you get

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Because is its own derivative, our antiderivative for the function will probably be very similar to the original function. To check our first antiderivate candidate we have to use the chain rule, , where and . Then and , so we calculate: Hence our candidate almost works, we just have to bring the minus sign over: . Using the chain rule to double check, we find that, indeed, This is one antiderivative. To find another one, we can simply add a constant to the antiderivative shown above, say . When we differentiate, the constant will disappear, giving us the same derivative as before. 