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Unformatted text preview: Elementary Antiderivatives for Exam Purposes • You are NOT required to show work for an antiderivative that is on this list. • If an antiderivative looks like one of these, but with something linear replacing x, you are still allowed to show no work. • For ANY OTHER ANTIDERIVATIVE, you must show all steps and I must be convinced that you did the work, not your calculator. 1. 2. 3. 4. 5. xr dx = xr+1 + C, r+1 if r = −1 6. 7. 8. 9. 10. ekx dx = 1 kx e +C k 1 dx = ln x + C x ex dx = ex + C sin x dx = − cos x + C cos x dx = sin x + C 1 sin kx dx = − cos kx + C k cos kx dx = 1 sin kx + C k 1 dx = ln kx + a + C kx + a k 1 x dx +C = tan−1 x2 + k 2 k k 11. 12. 13. 14. 15. ln x dx = x ln x − x + C tan x dx = − ln  cos x + C cot x dx = ln  sin x + C sec x dx = ln  sec x + tan x + C csc x dx = − ln  csc x + cot x + C 16. 17. 18. 19. 20. sec2 x dx = tan x + C csc2 x dx = − cot x + C sec x tan x dx = sec x + C csc x cot x dx = − csc x + C √ dx = sin−1 x + C 2 1−x Other (possibly) Useful Formulas • I will not explain these formulas or how to use use them. • It is your job to know what the symbols in each formula mean • It is your job to know how, when, and why to apply any given formula. sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x 11 sin2 x = − cos 2x 22 11 cos2 x = + cos 2x 22 1 sin ax sin bx = [cos(a − b)x − cos(a + b)x] 2 1 sin ax cos bx = [sin(a − b)x + sin(a + b)x] 2 1 cos ax cos bx = [cos(a − b)x + cos(a + b)x] 2 ∆x S= (y 0 + 4 y 1 + 2 y 2 + 4 y 3 + · · · + 2 y n − 2 + 4 y n − 1 + y n ) 3 M (b − a)3 ET  ≤ 12n2 M (b − a)5 ES  ≤ 180n4 ...
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 Fall '08
 STAFF
 Calculus, Antiderivatives, Derivative

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