Iнтеграл
Д/з
1)
8
Z
1√x+1
3
√x2
dx =
8
Z
1x+ 2 6
√x+1
3
√x2dx = x2
2+12 6
√x7
7+ 3 3
√x!
8
1
= 32 + 96√2
7+ 6 −
1
2−
12
7−3 =
=3
14 64√2 + 153
2)
π
2
Z
0
cos2(x)dx =
π
2
Z
0
1 + cos(2x)
2dx =x
2+1
4sin(2x)
π
2
0
=π
4
Iндивiдуальне завдання
1)
Z6x5+ sin(4x) + 3
cos2(x)dx =Z6x5dx +Zsin(4x)dx +Z3
cos2(x)dx =x6
−
1
4cos(4x) + 3 tg(x) + C
2)
Zparctg(x)
x2+ 1 dx =
t= arctg(x)
dt =dx
x2+ 1
=Z√t dt =2
3
2
√t3+C=2
3
2
qarctg3(x) + C
3)
2
Z
0
(x−1)e2xdx =
u=x−1du =dx
dv =e2xv=e2x
2
=(x−1)e2x
2
2
0−
2
Z
0
e2x
2dx =e4+ 1
2−e2x
4
2
0
=
=e4+ 1
2−
e4
−1
4=e4+ 3
4
1
