[ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx A p.2/29

A p./29

[ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx A p.2/29

[ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx [ ] F(x) f(x) C F(x) + C f(x) A p.2/29

[ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx [ ] F(x) f(x) C F(x) + C f(x) f(x)dx f(x)dx = F(x) + C ( C ) A p.2/29

[ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx [ ] F(x) f(x) C F(x) + C f(x) f(x)dx f(x)dx = F(x) + C ( C ) [ ] x α dx = α + xα+ + C (α ) A p.2/29

A p.3/29

[ ] x α dx = xα+ α + + C (α ) dx = log x + C x A p.4/29

[ ] x α dx = xα+ + C (α ) α + e x dx = e x + C dx = log x + C x A p.4/29

[ ] x α dx = xα+ + C (α ) α + e x dx = e x + C sin xdx = cosx + C dx = log x + C x cosxdx = sin x + C A p.4/29

[ ] x α dx = xα+ + C (α ) α + e x dx = e x + C sin xdx = cosx + C dx = tanx + C cos 2 x dx = log x + C x cosxdx = sin x + C sin 2 x dx = tanx + C A p.4/29

[ ] x α dx = xα+ + C (α ) α + e x dx = e x + C sin xdx = cosx + C dx = tanx + C cos 2 x x 2 + dx = tan x + C dx = log x + C x cosxdx = sin x + C sin 2 x dx = tanx + C A p.4/29

[ ] x α dx = xα+ + C (α ) α + e x dx = e x + C sin xdx = cosx + C dx = tanx + C cos 2 x x 2 + dx = tan x + C x 2 dx = sin x + C dx = log x + C x cosxdx = sin x + C sin 2 x dx = tanx + C x 2 dx = cos x + C A p.4/29

A p.5/29

[ ] αf(x)dx = α f(x)dx ( α ) A p.6/29

[ ] αf(x)dx = α f(x)dx ( α ) {f(x) + g(x)} dx = f(x)dx + g(x)dx A p.6/29

[ ] αf(x)dx = α f(x)dx ( α ) {f(x) + g(x)} dx = f(x)dx + f(g(x))g (x)dx = F(g(x)) + C g(x)dx ( F(y) f(y) ) A p.6/29

[ ] αf(x)dx = α f(x)dx ( α ) {f(x) + g(x)} dx = f(x)dx + f(g(x))g (x)dx = F(g(x)) + C g(x)dx ( F(y) f(y) ) [ ] A p.6/29

A p.7/29

(ax + b) = a F(x) f(x) f(ax + b)dx = f(ax + b)(ax + b) dx = F(ax + b)+c a a (ax + b) α dx = (ax + b)α+ + C (α ) a α + A p.8/29

(ax + b) = a F(x) f(x) f(ax + b)dx = f(ax + b)(ax + b) dx = F(ax + b)+c a a (ax + b) α dx = (ax + b)α+ + C (α ) a α + (cosx) = sin x f(cosx) sin xdx = F(cosx)+C tan xdx = log cos x + C A p.8/29

(ax + b) = a F(x) f(x) f(ax + b)dx = f(ax + b)(ax + b) dx = F(ax + b)+c a a (ax + b) α dx = (ax + b)α+ + C (α ) a α + (cosx) = sin x f(cosx) sin xdx = F(cosx)+C tan xdx = log cos x + C ( x ) a + b = a ( x ) ( x ) ( x ) ( x ) f a +b dx = a f a +b a +b dx = af a +b +C x 2 + a 2dx = a 2 ( x 2 dx = a) + a tan x a + C A p.8/29

A p.9/29

[ ] (i) sin(ax + b)dx (a 0) (ii) (iii) (iv) (v) a x dx (a > 0) (a > 0) a2 x2dx f(tanx) x 2 a 2dx cos 2 x dx A p.0/29

[ ] (i) a cos(ax + b) + C A p./29

[ ] (i) a cos(ax + b) + C (ii) a x log a + C A p./29

[ ] (i) a cos(ax + b) + C (ii) a x log a + C (iii) sin x a + C A p./29

[ ] (i) a cos(ax + b) + C (ii) a x log a + C (iii) sin x a + C (iv) F(tanx) + C F (x) = f(x) A p./29

[ ] (i) a cos(ax + b) + C (ii) a x log a + C (iii) sin x a + C (iv) F(tanx) + C (v) 2a log x a x + a F (x) = f(x) + C (a > 0) A p./29

A p.2/29

[ ] f (x)g(x)dx = f(x)g(x) f(x)g (x)dx A p.3/29

[ ] f (x)g(x)dx = f(x)g(x) f(x)g (x)dx (f(x)g(x)) = f (x)g(x) + f(x)g (x) A p.3/29

[ ] f (x)g(x)dx = f(x)g(x) f(x)g (x)dx (f(x)g(x)) = f (x)g(x) + f(x)g (x) [ ] log xdx = (x) log xdx A p.3/29

[ ] f (x)g(x)dx = f(x)g(x) f(x)g (x)dx (f(x)g(x)) = f (x)g(x) + f(x)g (x) [ ] log xdx = (x) log xdx = x log x x x dx = x log x dx = x log x x + C A p.3/29

A p.4/29

[ ] n 2 (A > 0 ) dx (x 2 +A) = ( ) x 2n 3 dx + n A (2n 2)(x 2 +A) n 2n 2 (x 2 +A) n A p.5/29

[ ] n 2 (A > 0 ) dx (x 2 +A) = ( ) x 2n 3 dx + n A (2n 2)(x 2 +A) n 2n 2 (x 2 +A) n [ ] A dx (x 2 +A) n = (x 2 +A) x 2 (x 2 + A) n dx A p.5/29

[ ] n 2 (A > 0 ) dx (x 2 +A) = ( ) x 2n 3 dx + n A (2n 2)(x 2 +A) n 2n 2 (x 2 +A) n [ ] dx (x 2 A (x 2 +A) = +A) x 2 n (x 2 + A) dx ( n dx = (x 2 +A) n n+ (x 2 +A) n ) x 2 dx A p.5/29

[ ] n 2 (A > 0 ) dx (x 2 +A) = ( ) x 2n 3 dx + n A (2n 2)(x 2 +A) n 2n 2 (x 2 +A) n [ ] dx (x 2 A (x 2 +A) = +A) x 2 n (x 2 + A) dx ( n ) dx = (x 2 +A) x n n+ (x 2 +A) n 2 dx dx = (x 2 +A) x n 2( n+)(x 2 +A) + dx n 2( n+)(x 2 +A) n A p.5/29

[ ] n 2 (A > 0 ) dx (x 2 +A) = ( ) x 2n 3 dx + n A (2n 2)(x 2 +A) n 2n 2 (x 2 +A) n [ ] dx (x 2 A (x 2 +A) = +A) x 2 n (x 2 + A) dx ( n ) dx = (x 2 +A) x n n+ (x 2 +A) n 2 dx dx = (x 2 +A) x n 2( n+)(x 2 +A) + dx n 2( n+)(x 2 +A) n x 3 dx = +2n (2n 2)(x 2 +A) n 2n 2 (x 2 +A) n A p.5/29

A p.6/29

[ ] ( ) ( ) A p.7/29

[ ] ( ) ( ) A p.7/29

[ ] ( ) ( ). A p.8/29

[ ] ( ) ( ). 2. (x a) n A p.9/29

[ ] ( ) ( ). 2. (x a) n 3. (x a) {(x a) 2 + b} n (b > 0) A p.20/29

[ ] ( ) ( ). 2. (x a) n 3. 4. (x a) {(x a) 2 + b} n (b > 0) {(x a) 2 + b} n (b > 0) A p.2/29

[ ] ( ) ( ). 2. (x a) n 3. 4. (x a) {(x a) 2 + b} n (b > 0) {(x a) 2 + b} n (b > 0) A p.22/29

A p.23/29

. A p.24/29

. 2. n + (n 2) log x a (x a) n A p.25/29

. 2. n + (n 2) log x a (x a) n 3. 2( n + ) (n 2) {(x a) 2 + b} n 2 log (x a)2 + b A p.26/29

. 2. n + (n 2) log x a (x a) n 3. 2( n + ) (n 2) {(x a) 2 + b} n 2 log (x a)2 + b 4. (x a) 2 + b dx = tan x a b b A p.27/29

. 2. n + (n 2) log x a (x a) n 3. 2( n + ) (n 2) {(x a) 2 + b} n 2 log (x a)2 + b 4. (x a) 2 + b dx = tan x a b b A p.28/29

38(75 ) 39(78 ) 42(83 46(92 ) A p.29/29