Differentiation and Integration Rules
Transcription
Differentiation and Integration Rules
Differentiation and Integration Rules BCCC Tutoring Center This handout highlights some of the most frequently encountered rules for differentiation and integration. For the following let u and v be functions of x, let n be an integer, and let a, c and C be constants. Fundamental Rules d(c) =0 dx R du d(cu) =c dx dx R d(u + v) du dv = + dx dx dx R R R v d( uv ) = dx du dx −u v2 cdu = c (du + dv) = du d(un ) = nun−1 dx dx du dv d(uv) =v +u dx dx dx du = u + C R un du = R du + C du + R dv + C un+1 +C n+1 u dv = uv − R v du + C dv dx Trigonmetric Functions d(sin u) = cos u dx R du dx d(cos u) du = − sin u dx dx sin u du = − cos u + C R cos u du = sin u + C d(tan u) du = sec2 u dx dx R tan u du = ln | sec u| + C d(cot u) du = − csc2 u dx dx R cot u du = ln | sin u| + C d(sec u) du = sec u tan u dx dx R sec u du = ln | sec u + tan u| + C Trigonmetric Functions Continued d(csc u) du = − csc u cot u dx dx R R R csc u du = − ln | csc u + cot u | + C R sec2 u du = tan u + C R sec u tan u du = sec u + C csc2 u du = − cot u + C csc u cot u du = − csc u + C Exponential and Logarithmic Functions d(eu ) du = eu dx dx R eu du = eu + C R 1 du = ln |u| + C u d(ln u) 1 du = dx u dx d(au ) du = (au )(ln a) dx dx R au du = au +C ln a v d(uv ) = uv (v 0 ln u + u0 ) dx u Inverse Trigonometric Functions d(sin−1 u) 1 du =√ 2 dx 1 − u dx Z d(cos−1 u) −1 du =√ dx 1 − u2 dx Z d(tan−1 u) 1 du = 2 dx 1 + u dx d(csc−1 u) −1 du = √ dx |u| u2 − 1 dx √ −1 u = − sin−1 + C 2 2 a a −u Z a2 d(cot−1 u) −1 du = dx 1 + u2 dx d(sec−1 u) 1 du = √ 2 dx |u| u − 1 dx 1 u √ = sin−1 + C a a2 − u 2 Z Z Z 1 1 u = tan−1 + C 2 +u a a −1 1 −1 u = − tan +C a2 + u 2 a a 1 1 u √ = sec−1 | | + C 2 2 a a |u| u − a 1 u −1 √ = − sec−1 | | + C a a |u| u2 − a2 Hyperbolic Trigonometric Functions d(sinh u) du = cosh u dx dx R sinh u du = cosh u + C du d(cosh u) = sinh u dx dx R cosh u du = sinh u + C d(tanh u) du = sech2 u dx dx R tanh u du = − ln |sech u | + C Inverse Hyperbolic Trigonometric Functions 1 du d(sinh−1 u) =√ dx 1 + u2 dx R √ 1 d(cosh−1 u) du =√ 2 dx u − 1 dx R √ 1 du d(tanh−1 u) = 2 dx 1 − u dx R 3 1 du = sinh−1 u + C 1 + u2 1 u2 −1 du = cosh−1 u + C 1 du = tanh−1 u + C 2 1−u