08 WFiIS Inf st 1 rok


(x2-1)3
2 5 x x
(5x2 - 6x + 3 - + )dx dx dx dx
x x2 x 1+x2 (x2+3)6
" " " " "
3 " "
3 4 4
x x+ x x x-x x
3+5 x2
" "
dx dx dx 3x + 5dx ax + bdx
3
x2 x
x3
"
x
2 2
e
2x x
" "6x
dx dx dx dx xe-x dx
3 5
x2
1
2x2+1 x3+3
x2-
6
"
"
tgx 1+ ctgx
x a2 - x2dx tgxdx ctgxdx dx sx
cos2 x
sin2 x
" "
3
tgx+3 cos x
cos x
"
dx dx dx sin5 x cos xdx
cos2 x x 1+4 sin2 x
ctgx
dx ln5 x
cos xesin xdx dx dx 61-xdx
x ln x x ln (sin x)
dx
arctgxdx arcsin xdx
x ln x ln (ln x)
arccos xdx x sin xdx xn ln xdx x3exdx
"
ln xdx ln2 x ex cos xdx x ln xdx
3 4
x
"
x3 ln2 xdx x2e-2x +1dx x3e2x +5dx dx
2x2+3
x3
"
dx x2exdx x ln2 xdx x cos xdx
4x4+1
"
x2 cos xdx x2 sin xdx 1 - x2dx x3xdx
"
"
x 5 ln x+7
dx x 4 - x2dx dx sin2 xdx
cos2x x
cos x 2x2
"
dx dx 5(sin x)9 cos xdx x7 cos x4dx
2 sin3 x
4x3+5
tgx
(2x + 3) ln xdx sin x cos xecos xdx etgxdx
(cos x)2
"
(ln x)3
ln x
x ln (9 + x2)dx (x2 + 3) sin 3xdx dx dx
x x
x2
dx
2+x3
dx dx 2x-1 x+2 2x-3
dx dx dx
(3x+5)6 2x+3 2x+1 3x+4 x2-3x+3
4x+3 1 5x dx dx
dx dx dx
2x2+3x+1 2x-3x2 2+3x 6x2-13x+6 -x2+6x-5
dx dx x+13 4x-5 x4
dx dx dx
x2-6x+13 2x2-2x+5 x2-4x-5 2x2-5x+3 x2+1
x+6 x+2 x+6 x+7 dx
dx dx dx dx
x+5 2x+3 3x+8 2x+9 3x2+3x
3x+5 6x-3
dx dx
x2+1 2x+1
7x2+7x-176 x3-4x2+1 3x2-5x+2 x4
dx dx dx dx
x3-9x2+6x+56 (x-2)4 x3-2x2+3x-6 x2+1
2x4-10x3+21x2-20x+5 17x2-x-26 2x 6x3+4x+1
dx dx dx dx
x2-3x+2 (x2-1)(x2-4) (x2+1)(x2+3) x4+x2
dx dx dx x3-2x2+5x-8
dx
x3+x2+x x4+x2+1 x4+64 x4-8x2+16
"
" "
3
x
1 dx 3
" " "
dx 2x + 1dx dx x x - 4dx
6 3
4x+3 3x-4
x+ x5
"
1+"x
x2 dx dx dx"
" " "
dx dx "
3 3
3 x+2 x x+a x x-a 1- x
x+2 x2
"
x-1
8x+3 dx dx
" " " " "x+3
dx dx dx
(x-1)2 x-2 4x2+3x+1 2x-x2 7-6x-x2 1-4x2
x x+1 dx dx
" " " "
dx dx
-3x2-2x+1 -x2+2x+8 x2+3x+2 4x2+3x-1
"
dx dx 3x+2
" " " "x+a
dx dx -x2 - 2x + 3dx
x2-x+m x2-5x+19 x2-ax
(x-a)(x-3a)
"
x2 x 2x2+3x+1
" "
x2 - 3x + 2dx dx dx dx
1-x
1-x2 x2+1
"
3
dx
"x -x+1 " "dx
dx x -x2 + x + 6dx
x2+2x+2 (x+1) x2-1 (x+1) 1+2x-3x2
dx
"
(x2+1) x2-1
sin (ax) cos (bx)dx sin7 xdx sin4 xdx sin4 x cos3 xdx
dx dx dx
sin2 x cos2 xdx x tgxdx
sin x cos sin x cos x
2
dx
ctgxdx sin2 xdx cos2 xdx tg2xdx
sin2 x cos2 x
cos3 x sin x
ctg2xdx tg4xdx ctg4xdx dx dx
cos2 x
sin4 x
cos(5x) cos(7x)dx sin(2x) sin(5x)dx sin(3x) cos(2x)dx sin3 xdx
sin4 xdx sin2(2x)dx cos5 xdx tg5xdx sin5 x cos2 xdx
cos x
"sin (2x)
sin3 x cos3 xdx dx sin xtgxdx dx
sin8 x
1+cos2 x
dx dx dx 3+sin2 x sin x cos x
dx dx
cos3 x sin x 5+4 cos x 2 cos2 x-cos4 x
sin3 x 1+sin4 x
cos7 xdx
arcsin x dx arccos x dx arctgx dx arcctgx dx x2arctgx dx
xarctgx arcsin(ex)
dx
"
dx dx x(1 + x2)arctgxdx
(1+9x2) arctg3x (1+x2)2 ex
dx
ln xdx logp xdx ln(x2 + 1)dx (ln x)2dx
x ln x
" "
dx dx ex-e-x
ln(x+ x2 + 1)dx (e3x+ ex)dx dx
3x+1 ex+e-x 2
ex-e-x dx ex ex
" "
dx dx dx
ex+e-x ex+5
3+2ex 3-5e2x
sin(2x) = 2 sin x cos x
5 5 x6 3 3
x3 - 3x2 + 3x - 2 ln |x| - + C - x4 + x2 - ln |x| + C
3 x 6 4 2
1 3 13 23
1 -1 4 6 12
ln (1 + x2) + C + C 3x3 - x- 4 6
+ C x - x12 + C
2 10(x2+3)5 3 13 23
1 1 3 3 2
2 2 3
-6x- 2
+ 30x6 + C (3x + 5)2 + C (ax + b)2 + C (2x2 + 1)3 + C
9 3a 4
4 1
2
1 5
x2 - + C (x2 + 3)5 + C -ex + C -1e-x + C
6 2 2
3
1
-1(a2 - x2)2 + C - ln | cos x| + C ln | sin x| + C tg2x + C
3 2
3 4 "
2 3
-ctgx - (ctgx)2 + C (tgx + 3)3 + C 2 sin x + C
3 4
1 1
arctg(2 sin x) + C sin6 x + C esin x + C ln | ln |x|| + C
2 6
1 -1
ln6 x + C ln | ln | sin x|| + C 61-x + C ln | ln | ln |x||| + C
6 ln 6
"
1
xarctgx - ln (1 + x2) + C xarctgx + 1 - x2 + C
2
"
xn+1 1
xarccosx - 1 - x2 + C sin x - x cos x + C (ln x - ) + C
n+1 n+1
(x3 - 3x2 + 6x - 6)ex + C x(ln x - 1) + C x(ln2 x - 2 ln x - 2) + C
"
1 2 2 x4 1 1
ex(sin x + cos x) + C x x(ln x - ) + C (ln2 x - ln x - ) + C
2 3 3 4 2 6
" "
3 4
1 1 1
-1e-2x +1 + C e2x +5 + C 2x2 + 3 + C 4x4 + 1 + C
6 8 2 8
x2 1
ex(x2 - 2x + 2) + C (ln2 x - ln x + ) + C x sin x + cos x + C
2 2
x2 sin x + 2x cos x - 2 sin x + C -x2 cos x + 2x sin x + 2 cos x + C
"
1 x x 1
arcsin x + 1 - x2 + C 3x - 3x + C xtgx - ln | cos x| + C
2 2 ln 3
ln2 3
2 x 1
-1 (4 - x2)3 + C (5 ln x + 7)3 + C - sin 2x + C
3 15 2 4
"
-1 1 1
+ C 4x3 + 2 + C sin10 x + C
3 2
4 sin2 x
1 x2
(x4 sin (x4) + cos (x4)) + C (x2 + 3x) ln x - - 3x + C
4 2
ecos x(1 - cos x) + C etgx(tgx - 1) + C
1 2 2
(9 + x2)[ln (9 + x2) - 1] + C -1(x2 + 3) cos 3x + x sin 3x + cos 3x + C
2 3 9 27
3
1 2 1
ln4 x + C (ln x)2 + C ln |x3 + 2| + C
4 3 3
1 1 1
-15 (3x+5)5 + C ln |2x + 3| + C x - ln |2x + 1| + C
2
x 2
+ ln |3x + 4| + C ln |x2 - 3x + 3| + C ln |2x2 + 3x + 1| + C
2 9
1 1 1 1
ln |x| - ln |2 - 3x| + C ln |3x + 2| + C ln |2x-3| + C
2 2 3 5 3x-2
3
1 1 x-3 1 2 1
ln |x-1| + C arctg + C arctg (x - ) + C ln |(x-5) | + C
4 x-5 2 2 3 3 2 (x+1)2
x3
ln |2x2 - 5x + 3| + C - x + arctgx + C x + ln |x + 5| + C
3
x 1 x 10 x 5
+ ln |2x + 3| + C + ln |3x + 8| + C + ln |2x + 9| + C
2 4 3 9 2 4
1 x 3
ln |x+1| + C ln (x2 + 1) + 5arctgx + C 3x - 3 ln |2x + 1| + C
3 2
7+2(x-2)+2(x-2)2
8 ln |x - 7| + 2 ln |x - 4| - 3 ln |x + 2| + C + ln |x - 2| + C
3(x-2)3
4 17 1 x x3
" "
ln |x - 2| + ln(x2 + 3) + arctg + C - x + arctgx + C
7 14 3
7 3 3
5
2 3 1 1
x3 - 2x2 + 5x + ln |x2 - 3x + 2| - ln |x-2| + C ln |(x-1) (x-2)10 | + C
3 2 2 x-1 3 (x+1)4(x+2)11
2
1 +1 1
ln(x ) + C ln(x6 + x4) - - arctgx + C
2 x2+3 x
"
2
x 3 2x+1 1 1 2x+1 2x-1
" " " "
ln |"x2+x+1| - arctg + C ln |x +x+1| + [arctg + arctg ] + C
3 4 x2-x+1
3 2 3 3 3
2
1 +4x+8 1 x+2 x-2 1 1 x
ln |x | + [arctg + arctg ] + C [ln(x2 + 4) - ] - arctg + C
128 x2-4x+8 64 2 2 2 x2+4 2
"
" " "
1 4x+3
3 6 6
3 x - 6 x + 6 ln | x + 1| + C x > 0 (2x + 1)3 + C + C
3 2
"
3
7 4 8 5 2
(3x+4)2
3 3 12
+C (x-4)3 +3(x-4)3 +C (x+2)3 - (x+2)3 +6(x+2)3 +C
2 7 8 5
" "
" "
1 2 x-a
"
ln |"x+a-"a "
| + C arctg + C -x - 4 x - ln |1 - x| + C
a
a x+a+ a a
x-1
"
1+
" " "
x-2
3 3 3
3 6 6
x - x + ln |2 x + 1| + C ln | | + C 2 ln | 4x2 + 3x + 1| + C
2 2 4 x-1
1-
x-2
"
x+3 3
arcsin (x - 1) + C arcsin + C -1 1 - 4x2 + arcsin (2x) + C
4 4 2
" "
1 3x+1 x-1
-3 -3x2 - 2x + 1 - arcsin + C 2 arcsin - -x2 + 2x + 8 + C
3 2 3
" "
1 3
ln |2x + 3 + 4x2 + 12x + 8| + C ln |2x + + 4x2 + 3x - 1| + C
2 4
"
1
ln |x - + x2 - x + m| + C ln |x - 2a + (x - a)(x - 3a)| + C
2
" " "
5 3 a
ln |x - + x2 - 5x + 19| + C x2 - ax + a ln |x - + x2 - ax| + C
2 2 2
"
x+1
(x + 1) -x2 - 2x + 3 + 4 arcsin + C
2
" " "
1 3 1 3 1 x
(x - ) x2 - 3x + 2 - ln |x - + x2 - 3x + 2| + C arcsin x - 1 - x2 + C
2 2 8 2 2 2
" " "
- x3 - 2x2 + 2x - arcsin 1 - x2 + C (x + 3) x2 + 1 + C
" "
1 5
(2x3 - 5x - 1) x2 + 2x + 2 + ln |x + 1 + x2 + 2x + 2| + C
6 2
51 25 2x-1 x-1 x-1
12(4x2 - x - ) + arcsin + C + C x > 1 - + C x < 1
2 4 5 x+1 x+1
" " "
2
2x 2
ą1 arcsin + C ln |"2x+"x -1| + C
2 x+1 4
2x- x2-1
1 1 3 1
-2(a+b) cos(a + b) - cos(a - b) + C - cos x + cos3 x - cos5 x - cos7 x + C
2(a-b) 5 7
3 3 1 1
-1 sin3 x cos x + x - sin(2x) + C sin5 x - sin7 x + C
4 8 16 5 7
x 1 1 x x
+ sin3 x cos x - sin(2x) + C ln |tg | + C ln |tg(Ą + )| + C
8 4 16 2 4 2
ln |tgx| + C - ln | cos x| + C ln | sin x| + C tgx - ctgx + C
x 1 x 1
- sin(2x) + C + sin(2x) + C tgx - x + C -ctgx - x + C
2 4 2 4
1 2
tg3x - tgx + x + C -1ctg3x + ctgx + x + C -1 cos2 x + + C
3 3 3 3 sin x
sin3 x
1 1 1 1 1
+ C sin(12)x + sin(2x) + C sin(3x) - sin(7x) + C
cos x 24 4 6 14
1 1 1
-10 cos(5x) - cos x + C cos3 x - cos x + C
2 3
3 3 x 1
- cos x sin3 x + x - sin(4x) + C - sin(4x) + C
8 32 2 8
2 1 1 1
sin x - sin2 x + sin5 x + C tg4x - tg2x - ln | cos x| + C
3 5 4 2
1 2 1 1
cos7 x + cos5 x - cos3 x + C cos4 x(1 cos2 x - ) + C
7 5 3 6 4
"
1 Ą
-7 sin7 x + C - sin x + ln |tg(x + )| + C -2 cos2 x + 1 + C
2 4
1 x 1 2 x
-2cos x + ln |tg | + C ln | sin x| + + C arctg(1tg ) + C
2 2 2 cos2 x 3 3 2
sin2 x
"
1 1
"
2tgx + arctg( 2tgx) + C arctg(2tg2x + 1) + C
2
2
3 1
sin x - sin3 x + sin5 x + sin7 x + C
5 7
" "
1
x arcsin x+ 1 - x2 +C x arccos x- 1 - x2 +C xarctgx- ln(1+x2)+C
2
"
1 x3 x2 1 2
xarcctgx + ln(1 + x2) + C arctgx - + ln(1 + x2) + C arctg3x + C
2 3 6 6 3
x
x+(x2-1)arctgx
1
+ C -arcsin (ex) + ln |tg(arcsin(e ))| + C
4(x2+1) 2ex 2 2
arctgx
x3 1
-x - + + C
4 12 4 cos2(arctgx)
x(ln x-1)
x(ln x-1)+C +C ln | ln x|+C x[ln(x2+1)-2]+2arctgx+C
ln p
" "
1 3x
x(ln2 x - 2 ln x + 2) + C x ln |x + x2 + 1| - x2 + 1 + C ln + C
ln 3 3x+1
"
e3x ex+e-x
+ 2 ex + C arctg(ex) + C + C ln(ex + e-x) + C
3 2
" "
x
1 1 5
" "2e +3-"3 "
ln | | + C ln(ex + 5) + C arcsin( ex) + C
3
3 2ex+3+ 3 5


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