Mathematics 1. (Integration)

Transcription

1 Mthemtics 1. (Integrtion) University of Debrecen fll

2 Definition Let I R be n open, non-empty intervl, f : I R be function. F : I R is primitive function of f if F is differentible nd F = f on I.

3 Definition Let I R be n open, non-empty intervl, f : I R be function. F : I R is primitive function of f if F is differentible nd F = f on I. If F, G : I R re primitive functions of f, then F G is constnt.

4 Definition Let I R be n open, non-empty intervl, f : I R be function. F : I R is primitive function of f if F is differentible nd F = f on I. If F, G : I R re primitive functions of f, then F G is constnt. Nottions for the primitive function f, or f (x)dx

5 Fundmentl primimtive functions: 1dx = x + c; x r dx = x r+1 r+1 + c; r 1 1 x = ln x + c; e x dx = e x + c; sin(x)dx = cos(x) + c; cos(x)dx = sin(x) + c; x dx = x ln dx, > 0 1 cos 2 (x) dx = tn(x) + c; 1 1+x 2 dx = rctn(x) + c

6 Let f, g : I R be functions nd, b R be constnts. Assume tht there exist primitives of f nd g, (f + bg) = f + b g.

7 Methods for integrtion: I.) If F is the primitive function of f, then for ll, b R 0, then f (x + b)dx = F (x + b) + c. Exmples () (2x 3) 10 dx = (2x 3) c; (b) 1 2 5x dx = (2 5x) c; 2 ( 5) (c) cos(4x 7)dx = sin(4x 7) 4 + c; (d) 3 2 x dx = 32 x ln(3)( 1) + c; (e) e 2x+1 dx = e 2x c.

8 Methods for integrtion: II.) If f : I R is differentible function, then f (x) dx = ln f (x) + c. f (x) Exmples () x+1 x 2 +2x 1 dx = 1 2 ln x 2 + 2x 1 + c; (b) x 2 x(x 4) dx = 1 2 ln x 2 4x + c; (c) 1 x ln x dx = ln ln(x) + c; (d) tn(x)dx = ln cos(x) + c; (e) e 2x 1+e dx = 1 2x 2 ln(1 + e2x ) + c.

9 Methods for integrtion: III.) If f : I R is differentible function, then for ll k 1 f k (x) f (x)dx = f k+1 (x) k c. If k = 1, then we cn pply Method II. Exmples () 2x(1 + x 2 ) 3 dx = (1+x 2 ) c; (b) x 2 (2x 3 + 4)dx = 1 (2x 3 +4) c; (c) sin 3 (x) cos(x)dx = sin4 (x) 4 + c; (d) 4x 3 + 2x 2 dx = (3+2x 2 ) c; (e) x (1+x 2 ) dx = 1 (1+x 2 ) c.

10 Methods for integrtion: IV.) If f : I R is differentible function, then e f (x) f (x)dx = e f (x) + c. Exmples () (1 + e x 1 )dx = x + e x 1 + c; (b) x e x 2 dx = 1 2 e x 2 + c.

11 (Integrtion by prts) Assume tht there exist primitives of f nd g. If dditionlly f nd g re differentible, then f (x)g(x)dx = f (x)g(x) f (x)g (x)dx. Exmples () e 2x (2x + 3)dx = e2x e2x 2 (2x + 3) 2 ; (b) e x (x 2 + 3)dx = e x (x 2 + 3) 2xe x + 2e x + c; (c) sin(x)(x + 14)dx = cos(x)(x + 14) + sin(x) + c; (d) ln(x)dx = x ln(x) x + c; (e) x 2 ln(x)dx = x 3 3 ln(x) x c.

12 (Chnge of vribles) Let g : I R be strictly monotone, differentible function. If there is primitive of f : g(i ) R, then there exists primitive of (f g)g nd ( ) f (x)dx g(x) = (f g)(x)g (x)dx. Exmples () sin(2x + 1)dx; (b) x cos(x 2 + 1)dx; (c) e 2x e x +1 dx.

13 Integrtion, Newton-Leibniz formul F (x) := x f (t)dt

14 Integrtion, Newton-Leibniz formul F (x) := x f (t)dt If f is Riemnn integrble, then F is continuous.

15 Integrtion, Newton-Leibniz formul F (x) := x f (t)dt If f is Riemnn integrble, then F is continuous. If f is continuous, then F is differentible, nd F (x) = f (x).

16 Integrtion, Newton-Leibniz formul F (x) := x f (t)dt If f is Riemnn integrble, then F is continuous. If f is continuous, then F is differentible, nd F (x) = f (x). (Fundmentl theorem of clculus) If f is continuous nd F is primitive function of f, then b f (x)dx = F (b) F ().

17 Integrtion, Newton-Leibniz formul F (x) := x f (t)dt If f is Riemnn integrble, then F is continuous. If f is continuous, then F is differentible, nd F (x) = f (x). (Fundmentl theorem of clculus) If f is continuous nd F is primitive function of f, then b f (x)dx = F (b) F (). It is enough to ssume tht f is Riemnn integrble, nd it hs differentible primitive function F.

18 Integrtion, Integrtion by prts, Chnge of vribles Let g : [A, B] R be strictly monotone incresing, continuously differentible function nd g(a) =, g(b) = b. If f : [, b] R is continuous, then b f (x)dx = B A f (g(y))g (y)dy.

19 Integrtion, Integrtion by prts, Chnge of vribles Let g : [A, B] R be strictly monotone incresing, continuously differentible function nd g(a) =, g(b) = b. If f : [, b] R is continuous, then b f (x)dx = B A f (g(y))g (y)dy. If g, f : [, b] R re differentible functions with integrble derivtives, then b f (x)g(x)dx = f (b)g(b) f ()g() b f (x)g (x)dx.