Wydział WILiŚ, Budownictwo i Transport, sem.1
dr Jolanta Dymkowska
Całka nieoznaczona
Zad.1 Oblicz:
"
" "
"
" 3
3 4 4
"
4 -1+x x-2 x x
3+5 x2
" "
1.1 2 x - x3 + 6x2 dx 1.2 dx 1.3 dx
x
x3

"
(1-x)2
4
"
1.4 dx 1.5 ( 2 + 3 x )2 dx 1.6 3x+1 - 3 ex dx
x x

1 2 1 sin 2x
1.7 5 sin x - "
dx 1.8 - dx 1.9 dx
cos2 x 1+x2 cos x
1-x2

2
1.10 tg x dx 1.11 e4x dx 1.12 3 sin 2x dx

1.13 cos 5x dx 1.14 sin2 x dx 1.15 cos4 x dx
" "
3
x2
"
1.16 3x + 1 dx 1.17 x 2x2 - 1 dx 1.18 dx
4
x3+1
1

2
x
e ln3 x
1.19 x e-x dx 1.20 dx 1.21 dx
x2 x

tg x
1.22 ctg x dx 1.23 sin5 x cos x dx 1.24 dx
cos2 x
"

3
tg x+3
cos x ex
"
1.25 dx 1.26 dx 1.27 dx
3ex+1 cos2 x
1+sin x

1 x 1
1.28 dx 1.29 dx 1.30 dx
(1+x2) arctg x x4+1 ex+e-x

2x ex 2x-3
"
1.31 dx 1.32 dx 1.33 dx
1+4x x2-3x+3
1-3e2x

1.34 x e-x dx 1.35 x2 cos 4x dx 1.36 3x2 - x + 2 3x dx

1.37 ln x dx 1.38 arctg x dx 1.39 arcsin x dx

2
ln2 x x
1.40 x3 e3x dx 1.41 dx 1.42 dx
x2 cos2 x

"
ln(arctg x)
2x+1
1.43 dx 1.44 x ln3 x dx 1.45 dx
sin2 x 1+x2

x2
"
1.46 ex sin 3x dx 1.47 sin(ln x) dx 1.48 dx
1-x2
Zad.2 Oblicz całki funkcji wymiernych:

3x-4 x+13 2x-1
2.1 dx 2.2 dx 2.3 dx
x2-x-6 x2-4x-5 x2-6x+9

1 3x+4 x
2.4 dx 2.5 dx 2.6 dx
3x2+2x+1 x2+4x+8 x3+1

x4-3x2-3x-2 2x3-19x2+58x-42 2x4-10x3+21x2-20x+5
2.7 dx 2.8 dx 2.9 dx
x3-x2-2x x2-8x+16 x3-3x+2

2x 6x3+4x+1 4x3+9x2+4x+1
2.10 dx 2.11 dx 2.12 dx
(x2+1)(x2+3) x4+x2 x4+3x3+3x2+x

1 3x2-17x+21 1
2.13 dx 2.14 dx 2.15 dx
x3(x-1)2(x+1) (x-2)3 x4+64
Zad.3 Oblicz całki funkcji trygonometrycznych:

3.1 cos 5x cos 7x dx 3.2 sin x cos 3x dx 3.3 cos5 x dx

3 4
3.4 tg x dx 3.5 ctg x dx 3.6 sin3 x cos4 x dx

cos3 x 1 1
3.7 dx 3.8 dx 3.9 dx
sin2 x sin3 x sin3 x cos x

1 1 1
3.10 dx 3.11 dx 3.12 dx
5+4 cos x 3+2 sin x 3 cos x+sin x+1

sin x cos x sin3 x sin 2x
"
3.13 dx 3.14 dx 3.15 dx
1+sin4 x 1+cos2 x
1+cos2 x
Zad.4 Oblicz całki funkcji niewymiernych:
" " "

3
x x x
" " "
4.1 dx 4.2 dx 4.3 dx
4 6
1+ x
3- x3 x+ x5
"
" "
3
x2- x+1
1 1-x
" "x+1+1
4.4 dx 4.5 � dx 4.6 dx
3
x1 x 1+x
x+1-1


2+x 1 1 x-1
"
4.7 dx 4.8 dx 4.9 � dx
2-x (x-1)2 x-2
(x+1) 1-x
"
1 2
" "
4.10 dx 4.11 dx 4.12 1 - x2 dx
4-x2 1-9x2