4 [ ] 4. ( ) (x ) 3 (x +). f(x) (x ) 3 x 3 3x +3x, g(x) x + f(x) g(x) f(x) (x 3)(x +)+x+ (x 3)g(x)+x+ g(x) r(x) x+ g(x) x (x+)+ x r(x)+ g(x) x x 4x+5 r(x) g(x) x (f(x) (x 3)g(x)) g(x) x f(x) f(x)g(x) 4 (x ) 3 (x +) x 4x+5 (x ) 3 x x + a x + a (x ) + a 3 (x ) a (x ) +a (x )+a 3 3 (x ) 3 x 4x+5 x a 3 x 4x+5 (x )(x 3) a (x )+a x 3 x a a 4 (x ) 3 (x +) x (x ) + (x ) x 3 x + C0. ( ) (4) () () x e x (3) log x (4) sin x x+ +x x C. ( ) 3x + x+ () () (3) (4) x x 3 +x (x +) x(x+) C. ( ) I n cos n xdx () I n () cos 4 xdx C3. ( ) f(x),g(x) f af(x), g (x) bg(x) (a,b ) (a b) f(x)g(x)dx f (x)g(x) f(x)g (x)+c (C ) C4. ( ) () e ax sinbx () e ax cosbx (3) (sinhax)sinbx
5 C0. ( ) (4) () () x e x (3) log x (4) sin x x+ +x x () C (x+ ) dx x dx x+ +x 3 (x+)3/ 3 x3/ +C ( (x+) 3/ x 3/) +C 3 () x e x dx x ( e x ) dx x e x (x ) ( e x )dx ( x e x + xe x dx x e x + xe x + ) e x dx x e x xe x e x +C (x +x+)e x +C dt (3) t log x dx x log x dx x t dt dx dx tdt t +C (log x ) (4) sin x (x) sin x sin xdx xsin x x(sin x) dx xsin x x dx x +C xsin x ( x ) dx xsin x+ x +C C. ( ) 3x + x+ () x () x 3 (3) +x (x +) (4) x(x+) ( ) () x ( x + ) +x x dx x + +x dx ( log x +log +x )+C log +x x +C
6 () 3x + (x 3 +x ) 3x + (x 3 x 3 +x dx +x ) x 3 +x dx log x3 +x +C (3) x (x +) dx + x (x +) dx x + x (x +) dx x + dx x tant dx dt cos t +tan t x + dx dx tan t+ dt dt dt t+c tan x+c +tan t (+tan t)dt x (x +) dx x x (x +) dx x x (x +) + tan x+c x + + x + dx (x +) dx ( x + x (x +) dx tan x x+ (x +) ) tan x +C ( ) x x + +tan x +C (4) (x+) x +x+ x(x+)+ (x+) (x+)x x(x+) x(x+) x x+ (x+) x+ x(x+) x+ (x+)(x+) x (x+) + x (x +x+)+x (x+) x x (x+)
7 x (x+) x (x+) (x+) (x+) x+ (x+) x+ x(x+) dx x x+ + (x+) dx log x log x+ x+ +C C. ( ) I n cos n xdx () I n () cos 4 xdx () n I n cos n xdx (sinx) cos n xdx sinxcos n x sinx (n )cos n x( sinx)dx sinxcos n x+(n ) sin xcos n xdx sinxcos n x+(n ) ( cos x)cos n xdx sinxcos n x+(n )(I n I n ) sinxcos n x+(n )I n (n )I n ni n sinxcos n x+(n )I n I n n sinxcosn x+ n n I n () I 0 I cos 0 xdx dx x+c cosxdx sinx+c
8 cos 4 xdx I 4 4 sinxcos3 x+ 3 4 I 4 sinxcos3 x+ 3 ( 4 sinxcosx+ ) I 0 4 sinxcos3 x+ 3 8 sinxcosx+ 3 8 x+c C3. ( ) f(x),g(x) f af(x), g (x) bg(x) (a,b ) (a b) f(x)g(x)dx f (x)g(x) f(x)g (x)+c (C ) (f (x)g(x) f(x)g (x)) f (x)g(x)+f (x)g (x) f (x)g (x) f(x)g (x) af(x)g(x) bf(x)g(x) (a b)f(x)g(x) f (x)g(x) f(x)g (x) (a b)f(x)g(x) (a b) f(x)g(x)dx f (x)g(x) f(x)g (x) ( ) C4. ( ) () e ax sinbx () e ax cosbx (3) (sinhax)sinbx () (e ax ) a e ax, (sinbx) b sinbx (a,b) (0,0) e ax sinbxdx a +b ((aeax )sinbx e ax (bcosbx))+c eax (asinbx bcosbx) a +b +C (a,b) (0,0) e ax sinbx 0 e ax sinbxdx 0dx C () (cosbx) b cosbx (a,b) (0,0) e ax cosbxdx eax (acosbx+bsinbx) a +b +C
9 (a,b) (0,0) e ax cosbx e ax cosbxdx dx x+c (3) (sinhax) a sinhax (a,b) (0,0) sinhaxsinbxdx acoshxsinbx bsinhaxcosbx a +b +C (a,b) (0,0) sinhaxsinbx 0 sinhaxsinbxdx 0dx C
30. Q(x) P(x) 0, Q(x) P(x). ( ) K R C f(x),g(x) K g(x) 0 K q(x),r(x) f(x) q(x)g(x)+r(x), (degr(x) < degg(x) r(x) 0) f(x), g(x) K degf(x) degg(x) q (x) r (x) f(x) q (x)g(x)+r (x), (degr (x) < degg(x) r 0) r (x) 0 g(x) f(x) q (x)g(x) r (x) 0 g(x),r (x) g(x) q (x)r (x)+r (x), (degr (x) < degr (x) r 0) q (x), r (x) r (x),r (x),...,r k (x) degg(x) > degr (x) > degr (x) > > degr k (x), r k (x) 0 r k (x) r k (x) r k (x) q k+ (x)r k (x)+r k+ (x), (degr k+ (x) < degr k (x) r k+ (x) 0) q k+ (x), r k+ (x) r k+ (x) degg(x) degr i (x) n degr n+ (x) 0 f(x) q (x)g(x)+r (x), g(x) q (x)r (x)+r (x), r (x) q 3 (x)r (x)+r 3 (x),. r n 3 (x) q n (x)r n (x)+r n (x), r n (x) q n (x)r n (x)+r n (x), r n (x) q n+ (x)r n (x) r n (x) r n (x) r n r n (x) r n (x) f(x), g(x) f(x),g(x) d(x) r (x) d(x) g(x),r (x) r (x) d(x) r(x)
3 r n (x) f(x), g(x) ( ) d(x) r n (x) f(x) g(x) r (x) f(x) q (x)g(x), r (x) g(x) q (x)r (x), r 3 (x) r (x) q 3 (x)r (x),. r n (x) r n 3 (x) q n (x)r n (x), r n (x) r n (x) q n (x)r n (x), d(x) r n (x) f(x), g(x) a(x), b(x) a(x)f(x)+b(x)g(x) d(x) f(x), g(x) ( ) a(x), b(x) a(x)f(x)+b(x)g(x). f(x),g(x) K f(x) g(x) d(x) K a(x),b(x) a(x)f(x)+b(x)g(x) d(x) f(x) g(x) a(x)f(x)+b(x)g(x) P(x) 0, Q(x) F(x) Q(x) P(x) F(x) () degq(x) degp(x) Q(x) g(x)p(x)+r(x), degr(x) < degp(x) R(x) 0 g(x),r(x) F(x) R(x) 0 F(x) Q(x) P(x) g(x)+ R(x) P(x)
3 degr(x) < degp(x) F(x) Q(x) degq(x) < degp(x) P(x) () F(x) P(x) P(x) a(x+α ) m...(x+α k ) m k (x +β x+γ ) n...(x +β l x+γ l ) n l f(x) (x+α ) m, g(x) a(x+α ) m...(x+α k ) m k (x +β x+γ ) n...(x +β l x+γ l ) n l a(x),b(x) ( ) a(x)f(x)+b(x)g(x) Q(x) P(x) Q(x)a(x)f(x)+Q(x)b(x)g(x) Q(x)a(x) + Q(x)b(x) f(x)g(x) g(x) f(x) a(x), b(x) (dega(x) < degg(x), degb(x) < degf(x)) Q(x) P(x) a(x) g(x) + b(x) f(x) b(x) (x+α ) m + a(x) g(x) g(x) Q(x) P(x) b (x) (x+α ) m + + b k (x) (x+α k ) m k + a (x) (x +β x+γ ) n + + a l (x) (x +β l x+γ l ) n l g(x) (degb i (x) < m i, dega j (x) < n j ) f(x) n (3) g(x) (degg(x) < ndegf(x)) degg(x) < degf(x) f(x) n degg(x) degf(x) g(x) q (x)f(x)+r (x), (degr (x) < degf(x)) degq (x) degf(x) q (x) f(x) q (x) q (x)f(x)+r (x), (degr (x) < degf(x))
33 degg(x) > degq (x) > degq (x) >... m < n g(x) q (x)f(x)+r (x), (degr (x) < degf(x)) q (x) q (x)f(x)+r (x), (degr (x) < degf(x)). q m (x) q m (x)f(x)+r m (x),(degr m (x) < degf(x)) q m (x) 0f(x)+r m+ (x),(degr m+ (x) < degf(x)) g(x) r m+ (x)f(x) m +r m (x)f(x) m + +r (x)f(x)+r (x) g(x) f(x) n r m+(x) f(x) n m + r m(x) f(x) n m+ + + r (x) f(x) n + r (x) f(x) n
34 [ ]. ( ) sinx dx. t tan x t dx sinx +t, dt +t sinx dx sinx dx +t dt dt t +t dt sinx sinx dx sin x dx tan t dt log t +C log x +C sinx cos x dx dt t cosx dx sinx sinx dx t dt ( t + ) +t (log t log +t )+C (log cosx log +cosx )+C log cosx +cosx +C C5. ( ) a < b I dx (x a)(b x) x a x a () t b x I tan b x +C (b a () ) (x a)(b x) ( x a+b ) x (a+b) I sin +C b a (3) C C C6. ( ) a, b () sin () (3) x cosx a+tanx acos x+bsin (a > 0,b 0) x C7. ( ) () x x+ 3 x () x + x (3) (4) x x 3 x + C8. ( ) logx () () (3) +e x +x x(+(logx) )
. C5. ( ) a < b I 35 (x a)(b x) dx x a x a () t b x I tan b x +C () (b a ) (x a)(b x) ( x a+b ) x (a+b) I sin +C b a (3) C C x a () t b x (b x)t (x a) x bt +a +t x a t b x x dt dx b x b a x a(b x) I (b x) dt dx (x a)(b x) b a b a (b x) (x a)(b x) dx dx x a +t dt tan +C tan b x +C () A x dx x sin A +C ) (b bt +a +t b a dt I (b a ) ( ) dx sin x a+b a+b x b a x a b +C sin +C b a x a (3) θ tan b x 0 θ < π tan θ x a b x cosθ tan (θ/) +tan (θ/) x (a+b) b a x a b x + x a b x a+b x b a ( π ) ( cosθ sin θ sin θ π )
36 π θ π < π x (a+b) sin θ π b a I θ +C θ π +C C C π C6. ( ) () sin () (3) x cosx a+tanx acos x+bsin (a > 0,b 0) x C () t tan x cosx t t dx +t, sinx +t, dt +t I sin x cosx dx ( ) t +t t +t dt +t +t t ( t )(+t ) dt +t t 4 +t dt +t ( ) (t + )(t ++ ) dt t + + t ++ dt ( ) t t+ + t + dt + 4 t log t+ + tan t +C + + 4 tan x log x tan x + + tan tan +C + + dt () t tanx dx cos x +t ( I a+tanx dx a+t+t dt +a a+t + a t ) dt +t +a a+t t +t + a +t dt +a +a ( log a+t ) log +t +atan t +C ( log a+tanx ) log +tan x +ax +C log acosx+sinx +ax +a +C
37 (3) t tanx cos x +t, sin x t dx +t dt +t I acos x+bsin x dx a +b t +t dt a+bt dt +t +t b > 0 I b t +(a/b) dt b ( b b a tan a t+c b tan )+C ab a tanx b < 0 I b t (a/( b)) dx ( ) b b a t a/( b) t+ dt a/( b) ab log t a/( b) t+ a/( b) +C ab log abtanx a +C abtanx+a C7. ( ) () x x+ () x + x (3) x x (4) 3 x 3 x + () x + t x x + t tx+x x t t x x + t x x x + dt dx dx x + dt x+ x + x+t x t I x x + dx t t dt t dt t log t x ++x t+ +C log x ++x+ +C t t+ dt dx x tanθ dθ cos θ I tanθ tan θ+cos θ dθ log cosθ cosθ + +C sinθ dθ sinθ cos θ dθ
38 cos θ cosθ +x I log +tan θ +x x + + x + θ < π cosθ > 0 +C log x + + x + +C log x + x ++ +C ( x ++x ) x ++(x ) x ++x x+ ( x ++(x ) ) x + x ) ( x + )( x ++x ( x ++x+) x ++(x+) x ++x +x+ ( x ++(x+) ) x ++x ) ( x ++)( x ++x x ++x log x ++x+ log x ++x x ++x+ log ( x + )( x ++x ) ( x ++ )( x ++x ) log x + x ++ x+ () t t ( x) x+ x t x +t dt dx x+ x ( x) ( x) x I x x dx x (+t ) (t ) +t dx x dt dx dx ( t ) +C t +t t +C ( (+t ) t dx (t ) +t ( ) (t ) dx ( t) + (+t) dx x+ x x +C x+ x x +t )
39 (3) t dt x+ dx x+ t x t x+ t I dx x t dt + t dt t+ t+ dt t+log t t t+ +C x++log x+ +C x++ (4) t 3 x x t 3 dx + dt 3t I 3 x 3 dx x + t 3 3 t +3t 3log t+ +C t t+ 3t dt 3 t t+ t+ dt x 3 (x )/3 +3(x ) /3 3log (x ) /3 + +C x 3 (x )/3 +3(x ) /3 3log (x ) /3 + +C C8. ( ) logx () () (3) +e x +x x(+(logx) ) () t e x dt dx ex t I dx +e x t +t dt C7() t ++t e x ++e x I log +C log t ++t+ e x ++e x + +C log e x + e x ++ +C () I logx dx +x +xlogx dx +x x
40 C7(3) I +xlogx 4 x+ log x+ +C x++ (3) t logx x e t dx dt et x(+(logx) ) dx e t (+t ) et dt +t dt tan t+c tan (logx)+c