201-NYB-05 - Calculus 2
WORKSHEET: INTEGRALS
1.
2.
3.
4.
5.
6.
7.
Z
Z
Z
Z
Z
Z
1
√
dx
(x − 2) x2 − 4x + 3
12.
arctan(1/x) dx
13.
√
15.
sin(sec(x)) sec2 (x) tan(x) dx
16.
x2 cot−1 (x) dx
−1
8.
14.
(2 sin(θ) + 1) cos(θ)
dθ
sin2 (θ) − sin(θ) − 6
Z 0
Z
es + 1 ds
e3x
√
dx
1 − e3x
sin(ln(x)) dx
17.
Z
√
2 x+4
√
dx
x(3 x + 1)
Z
3x2 + 8x + 4
dx
x2 (x + 1)2
Z
√
Z
sin(5x) cos(3x) dx
Z
3x2
dx
(x + 1)3
Z
sin3 (6x) dx
√
18.
19.
Z
9.
10.
11.
(x2
Z
1
dx
+ 1)5/2
w4 + 1
dw
w2 + 1
Z 9 q√
0
x + 1 dx
20.
21.
22.
9 − 4x2
dx
x2
Z π/3
π/6
Z
1
dx
2x − x2
cot(x)
dx
sin3 (x)
Z
ln(t2 + 9) dt
Z
sec 2 (x) tan3 (x) dx
Z
ln(x) + 1
3
x
q
3
ln(x)
dx
ANSWERS:
Note: the integration techniques suggested below only indicate one way to do each integral. You
may find others ways, maybe even better ways, to solve each problem.
1. complete the square + substitution
sec−1 (x − 2) + C
2. integrate by parts + substitution
1
1
x tan−1 ( ) + ln(x2 + 1) + C
x
2
√
3. substitution u = es + 1 + long division + partial fractions
√
es + 1 − 1 √
s
2 e + 1 + ln √ s
+C
e + 1 + 1
4. substitution u = sin θ + partial fractions
1
[7 ln | sin θ − 3| + 3 ln | sin θ + 2|] + C
5
5. substitution + integrate by parts
− sec(x) cos(sec(x)) + sin(sec(x)) + C
6. integrate by parts + long division + substitution
x6 1
1 3
x cot−1 (x) +
− ln(x2 + 1) + C
3
6
6
7. substitution u = 1 − e3x
2p
1 − e−3
3
8. integrate by parts + boomerang
x
(sin(ln(x)) − cos(ln(x))) + C
2
9. trig substitution
3
x
1
x
√
√
−
+C
1 + x2 3
1 + x2
10. long division
w3
− w + 2 tan−1 (w) + C
3
11. substitution u =
232
15
√
x+1
√
12. substitution u = x + partial fractions
√
√
20
8 ln( x) −
ln(3 x + 1) + C
3
13. partial fractions
4
1
− +
+C
x x+1
14. complete the square + substitution
sin−1 (x − 1) + C
15. integrate by parts + boomerang
1
− (3 sin(5x) sin(3x) + 5 cos(5x) cos(3x)) + C
16
16. substitution
u=x+1
1
2
+C
−
3 ln |x + 1| +
x + 1 2(x + 1)2
17. substitution u = cos(6x)
1
6
!
cos3 (6x)
− cos(6x) + C
3
18. trig substitution x = 32 sin θ
√
2x
9 − 4x2
− 2 sin−1
+C
−
x
3
19. substitution u = sin(x)
8
1
1 − 3/2
3
3
20. integrate by parts + long division
t
+C
t ln(t2 + 9) − 2t + 6 tan−1
3
21. substitution u = sec(x)
sec7/2 (x) sec3/2 (x)
2
−
7
3
!
+C
22. substitution u = ln(x)
ln5/3 (x) ln3/2 (x)
+
3
5
2
!
+C