Differentiation Formulas

Differentiation Formulas ′ ′ 1. (f (x) ± g(x)) = f ′ (x) ± g ′ (x) ′ 3. (f (x)g(x)) = f (x)g ′ (x) + g(x)f ′ (x) ′ 5. (f (g(x))) = f ′ (g(x)) · g ′ (x) d x (e ) = ex 8. dx d 10. (sin x) = cos x 11. dx 1 d (arcsin x) = √ 13. dx 1 − x2 7. d x (a ) = ax ln a (a > 0) dx d (cos x) = − sin x dx 2. (kf (x)) = kf ′ (x) ′  g(x)f ′ (x) − f (x)g ′ (x) f (x) = 4. 2 g(x) (g(x)) d n 6. (x ) = nxn−1 dx d 1 9. (ln x) = dx x d 1 12. (tan x) = dx cos2 x 1 d (arctan x) = 14. dx 1 + x2 A Short Table of Indefinite Integrals I. Basic Functions 1 1. x dx = xn+1 + C, n 6= −1 n+1 Z 1 dx = ln |x| + C 2. x Z 1 x 3. ax dx = a + C, a > 0 ln a Z 4. ln x dx = x ln x − x + C Z n 5. Z sin x dx = − cos x + C 6. Z cos x dx = sin x + C 7. Z tan x dx = − ln | cos x| + C II. Products of ex , cos x, and sin x 8. Z 9. Z 10. Z 11. Z 12. Z 1 eax [a sin(bx) − b cos(bx)] + C a2 + b2 1 eax [a cos(bx) + b sin(bx)] + C eax cos(bx) dx = 2 a + b2 1 sin(ax) sin(bx) dx = 2 [a cos(ax) sin(bx) − b sin(ax) cos(bx)] + C, a 6= b b − a2 1 [b cos(ax) sin(bx) − a sin(ax) cos(bx)] + C, a = 6 b cos(ax) cos(bx) dx = 2 b − a2 1 sin(ax) cos(bx) dx = 2 [b sin(ax) sin(bx) + a cos(ax) cos(bx)] + C, a 6= b b − a2 eax sin(bx) dx = III. Product of Polynomial p(x) with ln x, ex , cos x, sin x 1 1 xn+1 ln x − xn+1 + C, n 6= −1 n+1 (n + 1)2 Z Z 1 1 ax ax p′ (x)eax dx 14. p(x)e dx = p(x)e − a a 1 1 1 = p(x)eax − 2 p′ (x)eax + 3 p′′ (x)eax − · · · a a a (+ − + − . . .) (signs alternate) 13. Z xn ln x dx = Z 1 1 p′ (x) cos ax dx p(x) sin ax dx = − p(x) cos ax + a a 1 1 1 = − p(x) cos ax + 2 p′ (x) sin ax + 3 p′′ (x) cos ax − · · · a a a (− + + − − + + . . .) (signs alternate in pairs after first term) Z Z 1 1 p′ (x) sin ax dx 16. p(x) cos ax dx = p(x) sin ax − a a 1 1 1 = p(x) sin ax + 2 p′ (x) cos ax − 3 p′′ (x) sin ax − · · · a a a (+ + − − + + − − . . .) (signs alternate in pairs) 15. Z IV. Integer Powers of sin x and cos x 17. Z 18. Z 19. Z 20. Z 21. Z 22. Z 23. Z Z 1 n−1 sinn−1 x cos x + sinn−2 x dx, n positive n n Z 1 n−1 n n−1 cos x dx = cos cosn−2 x dx, n positive x sin x + n n Z m−2 −1 cos x 1 1 + dx, m 6= 1, m positive dx = sinm x m − 1 sinm−1 x m − 1 sinm−2 x 1 1 (cos x) − 1 dx = ln +C sin x 2 (cos x) + 1 Z 1 sin x m−2 1 1 dx = + dx, m 6= 1, m positive m m−1 m−2 cos x m − 1 cos x m−1 cos x 1 1 (sin x) + 1 dx = ln +C cos x 2 (sin x) − 1 sinn x dx = − sinm x cosn x dx: If m is odd, let w = cos x. If n is odd, let w = sin x. If both m and n are even and positive, convert all to sin x or all to cos x (using sin2 x + cos2 x = 1), and use IV-17 or IV-18. If m and n are even and one of them is negative, convert to whichever function is in the denominator and use IV-19 or IV-21. If both m and n are even and negative, substitute w = tan x, which converts the integrand to a rational function that can be integrated by the method of partial fractions. V. Quadratic in the Denominator 1 x 1 dx = arctan + C, a 6= 0 x2 + a2 a a Z bx + c b c x 25. dx = ln |x2 + a2 | + arctan + C, a 6= 0 2 2 x +a 2 a a Z 1 1 dx = (ln |x − a| − ln |x − b|) + C, a 6= b 26. (x − a)(x − b) a−b Z cx + d 1 27. dx = [(ac + d) ln |x − a| − (bc + d) ln |x − b|] + C, (x − a)(x − b) a−b 24. Z VI. Integrands Involving √ a2 + x2 , √ √ a2 − x2 , x2 − a2 , a>0 1 x dx = arcsin + C a a2 − x2 Z p 1 √ dx = ln x + x2 ± a2 + C 29. x2 ± a2   p Z Z p 1 1 2 2 2 2 2 √ dx + C a ± x dx = x a ±x +a 30. 2 a2 ± x2  p  Z p Z 1 1 2 2 2 2 2 √ 31. x x −a −a dx + C x − a dx = 2 x2 − a2 28. Z √ a 6= b