10.

25 2. ln(1 +p

2)

3. 1 2014 4. 2p

3

5. a), c)

6.

 1 3

2 7. 5

6 8. 1 p

3 3p

2⇡

9. 1 3

⇣5p 5 8⌘

10. 2e ^⇡/2

11. Ä Ñ

Z x⁵ 2x⁴+ 2x³ 4x²+ 5x x⁴+ 2x²+ 1 dx =

Z

x 2 + 4x + 2 (x²+ 1)²dx t‰. Ï0⌧,

Z 4x

(x²+ 1)²dx =

Z 2· 2x

(x²+ 1)²dx = 2

x²+ 1 + C1

t‡

Z 2

(x²+ 1)²dx =

Z 2 sec²✓

(tan²✓ + 1)²d✓ (x = tan ✓ \ XX)

= Z

2 cos²✓d✓ = Z

1 + cos 2✓d✓

= ✓ +1

2sin 2✓ + C2 = ✓ + sin ✓ cos ✓ + C2

= arctan(x) + x

x²+ 1 + C2

t¿\,

Z x⁵ 2x⁴+ 2x³ 4x²+ 5x

x⁴+ 2x²+ 1 dx = 1

2x² 2x + arctan(x) + x 2 x²+ 1 + C

2

sin ✓ = 2u

1 + u², cos ✓ = 1 u²

1 + u², d✓ = 2 1 + u²du t‰.

✓ = ⇡

3| L, u = 1 p3t‡,

✓ = ⇡

2| L, u = 1t¿\

Z ⇡/2

⇡/3

1

3 + 2 sin ✓ cos ✓d✓ = Z 1

1/p 3

✓ 1

3 + 2· 2u 1 + u²

1 u² 1 + u²

◆ · 2 1 + u²du

= Z 1

1/p 3

2

4u²+ 4u + 2du = Z 1

1/p 3

2

(2u + 1)²+ 1du

= arctan(2u + 1) ¹

1/p 3

= arctan(3) arctan

✓ 2 p3 + 1

◆

t‰.

13.

ln

✓1 + x 1 x

◆

= ln(1 + x) ln(1 x)

= X1 n=1

( 1)^{n 1} n xⁿ+

X1 n=1

1 nxⁿ

= X1 n=1

( 1)^{n 1}+ 1

n xⁿ

t¿\

an = 8>

><

>>

:

0 n@ ›⇠

2

n n@ @⇠

t‰.

0 ansin(n)

n  2

n² t‡, X1 n=1

2

n²@ ⇠4X¿\, X1

n=1

ansin(n)

n @ ⇠4\‰. (DP ⇣ ï)

X1 a sin(n) X¹ a sin(n)

14. (a)

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-2 -1.5 -1 -0.5 0 0.5 1

r=1

(b) 2 sin 2✓ = 1–⌧

1¨Ñt–⌧ P · t Ãòî P⇣X ✓î ⇡ 12, 5⇡

12t‰.

0|⌧, lXî ◆tî

A = 4

"Z 5⇡/12

⇡/12

1

2(2 sin 2✓)²d✓ 1 2

✓5⇡

12

⇡ 12

◆#

= 4

Z 5⇡/12

⇡/12

2 sin²(2✓)d✓ 2⇡

3

= 4

Z 5⇡/12

⇡/12

1 cos(4✓)d✓ 2⇡

3

= 4

✓5⇡

12

⇡ 12

◆

sin(4✓)^5⇡/12

⇡/12

2⇡

3

= p 3 + 2⇡

3

t‰.

15. x⁰(t) = sin t + csc t, y⁰(t) = cos t t¿\

ds = p

x⁰(t)²+ y⁰(t)²dt

= p

sin²t 2 + csc²t + cos²t dt =p

csc²t 1dt

= | cot t|dt

t‰.

0|⌧, lXî · X 8tî

s =

Z ⇡/3

⇡/6

| cot t|dt

= ln| sin t| ^⇡/3

⇡/6

= ln(p 3)