1.
p
y = sin x 2
=
cos x 1 + sin x dx = dy = cos x dx
p
Rp
2
1+t2
=
1 + y 2 dy = 1 + y 2 = y + t y = 1−t
=
dy
=
−
dt
2
2t
2t
R (1+t2 )2
R t
1
=−
dt = − 4 + 2t
+ 4t13 dt =
3
√
√ 4t
ln | 1+sin2 x−sin x|
( 1+sin2 x−sin x)2
−
+
=−
8
2
1
+ √
+C
R
8(
1+sin2 x−sin x)2
2.
R
dx
1+ex/2 +ex/3 +ex/6
=
R
=
6 dy
y(1+y+y 2 +y 3 )
R
6 dy
y
6 dy
y
R
=
R
= y = ex/6
−
= x − 3 ln(ex/6 + 1)
−
R
=
x = 6 ln y
6 dy
y(y+1)(y 2 +1)
R
3 dy
y+1
dx = y6 dy =
=
− 3y+3
y 2 +1 dy =
R 2y dy R 3 dy
3
2
y 2 +1 −
y 2 +1
3 dy
y+1 −
− 32 ln(ex/3
R
=
+ 1) − 3 arctan(ex/6 ) + C
3.
R
√
x−√x2 +3x+2
dx
x+ x2 +3x+2
=
√
2
2−t2
+6t−4 dx = −2t
dt
= x2 + 3x + 2 = x + t x = 2t−3
=
2
(2t−3)
R −t −2t2 +6t−4
R −2t3 +6t2 −4t
= 4−3t (2t−3)2 dt = (2t−3)(3t−4) dt =
2t−3
R
R
1
17t−12
= 18
−6t + 1 dt + (2t−3)(3t−4)
dt =
R
R 27 dt R 32 dt 1
−6t + 1 dt + 2t−3
− 3t−4 =
= 18
R
R 2 dt
R 3 dt 1
32
= 18
−6t + 1 dt + 27
2
2t−3 − 3
3t−4 =
√
2
= − (6x+1)(x+1− 18x +3x+2)−2x +
√
+ 34 ln |2 x2 + 3x + 2 − 2x − 3|−
√
16
− 27
ln |3 x2 + 3x + 2 − 3x − 4| + C
1