Integrals_tipus (2009)

Apunte Español
Universidad Universidad Autónoma de Barcelona (UAB)
Grado Ciencias Ambientales - 1º curso
Asignatura Mates
Año del apunte 2009
Páginas 2
Fecha de subida 25/05/2014
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Matem` atiques Ci` encies Ambientals Teoria LLISTA D’INTEGRALS IMMEDIATES 1.
dx = x + C.
xn+1 + C, n = −1.
n+1 2.
xn dx = 3.
dx = ln |x| + C.
x 4.
dx = ln |x + a| + C.
x+a 5.
√ dx √ = x + C.
2 x 6.
ax dx = ax + C; en particular, ln a ex dx = ex + C (perqu`e ln e = 1).
7.
sin xdx = − cos x + C.
8.
cos xdx = sin x + C.
9.
tan xdx = sin x dx = − ln cos x + C.
cos x 10.
cot xdx = cos x dx − ln sin x + C.
sin x 11.
dx = cos2 x (1 + tan2 x)dx = tan x + C.
12.
dx = sin2 x (1 + cot2 x)dx = cot x + C.
13.
√ 14.
dx = arctan x + C.
1 + x2 dx = arcsin x + C.
1 − x2 Tamb´e s´on integrals immediates les que veieu a continuaci´o i que s´on, en certa manera, “generalitzacions” de la llista que acabeu de llegir: 1.
f (x)dx = f (x) + C.
f (x)n+1 + C, n = −1.
n+1 2.
f (x)n f (x)dx = 3.
f (x) dx = ln |f (x)| + C.
f (x) 1 4.
5.
6.
f (x) dx = ln |f (x) + a| + C.
f (x) + a f (x) 2 f (x) dx = af (x) f (x)dx = f (x) + C.
af (x) + C; en particular, ln a 7.
f (x) sin f (x)dx = − cos f (x) + C.
8.
f (x) cos f (x)dx = sin f (x) + C.
9.
f (x) tan f (x)dx = − ln cos f (x) + C.
10.
f (x) cot f (x)dx = − ln sin f (x) + C.
ef (x) f (x)dx = ef (x) + C (perqu`e ln e = 1).
11.
f (x) dx = cos2 f (x) f (x)(1 + tan2 f (x))dx = tan f (x) + C.
12.
f (x) dx = sin2 f (x) f (x)(1 + cot2 f (x))dx = cot f (x)x + C.
13.
14.
f (x) 1 − f (x)2 dx = arcsin f (x) + C.
f (x) dx = arctan f (x) + C.
1 + f (x)2 EXEMPLES 1 x1/3 x x−2/3 dx = √ + C = 3 3 + C.
3 8 8 1/3 (a) dx √ = 3 8x2 (b) 5x x2 + 7dx = (c) 2x + 1 dx = ln |x2 + x − 1| + C.
x2 + x − 1 (d) √ (e) e−5x dx = (f) √ (g) dx = 4 + 9x2 1 (8x2 )−1/3 dx = √ 3 8 5 2 x 2 dx = −16 1 − 8x2 1 −5 (2x)(x2 + 7)1/2 = 5 (x2 + 7)3/2 5 + C = (x2 + 7) x2 + 7 + C.
2 3/2 3 −16x 2 √ =− 16 2 1 − 8x2 1 − 8x2 + C.
1 −5e−5x dx = − e−5x + C.
5 x dx = 1 − 4x4 x 1 − (2x2 )2 dx 1 9 2 = 4 4(1 + 4 x ) dx = 1 4 4x 1 − (2x2 )2 dx 2 3x 2 = 4 · 3 1+( 2 ) 2 dx = 1 arcsin(2x2 ) + C.
4 3/2 1 3x 3x 2 = 6 arctan( 2 ) + C.
1+( 2 ) ...