1 Antiderivatives of sec2 x and √ 1 − x2 � Example: sec2 x dx Searching for antiderivatives will help you remember the speciﬁc formulas d for derivatives. In this case, you need to remember that dx tan x = sec2 x. � sec2 x dx = tan x + c � 1 √ dx 1 − x2 � dx An alternate way to write this integral is √1−x . This is consistent with 2 the idea that dx is an inﬁnitesimal quantity which can be treated like any other number. d 1 We remember dx sin−1 x = √1−x and conclude that: 2 � dx √ = sin−1 x + c. 1 − x2 � dx Example: 1 + x2 � dx = tan−1 x + c 1 + x2 Example: When looking for antiderivatives, you’ll spend a lot of time thinking about derivatives. For a little while you may get the two mixed up and diﬀerentiate where you were meant to integrate, or vice-versa. With practice, this problem goes away. Here is a list of the antiderivatives presented in this lecture: � 1. sin x dx = − cos x + c where c is any constant. � 2. � 3. � 4. � xn dx = xn+1 +c n+1 dx = ln |x| + c x for n �= −1. (This takes care of the exceptional case n = −1 in 2.) sec2 x dx = tan x + c. dx √ = sin−1 x + c (where sin−1 x denotes “inverse sine” or arcsin, 1 − x2 1 and not .) sin x � dx 6. = tan−1 (x) + c. 1 + x2 5.

Antiderivatives of sec^2 x and 1/√1 - x^2

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