TUGAS KELOMPOK

KALKULUS 10

(KALKULUS I S/D KALKULUS VII)

v

NAMA / NIM :

Danang adiluhung 04211038

Firmansyah 04111042

A.

uv =

d

1.

u = x → dv = dx

dv = cos x dx → v =

= x snix + cosx + C

2.

a =

dv = 2 x dx → v =

=

3. a = x → du = dx

dv =

=

=

2 –

4.

a =

dv = cos x dx → v =

5.

=

=

6.

u =

dv = snixdx → v =

= - +

7.

2

Soal

1.

u = x → du = dx

dv = snix dx → v =

= - cosx

2.

u = x → du = dx

dv = sni2x dx

w = 2x

dw = 2dx

dx =

dv = sniw

=

=

= -

= x

= -

3.

u = enx → du=

dv = 1 dx → v =

=

=

= x

4.

u = x → du = dx

dv =

= -

=

= e

= x

= -

5.

dv =

=

-

=

=

=

6.

dv = snix dx → v =

= - cos x

=

u =

dv = conx dx → v =

= sinx

2

7.

a = arc sinx → du = dx

dv = dx → v =

= x

Atau →x dx = -

=

=

= -

= -

=

-

=

8.

u = arc tan

dv = dx → v =

= x

=

du = 2x dx

xdx =

=

ln

9.

Misal u =

dv =x dx

=

Uv =

=

=

=

dv =

= ln x

2

10. 1.

u = x-2 → du = dx

=

2.

3.

lnx +

lnx +

4.

=

= -cos x –

= -cos x –

= -cos x -

5.

Hal 59

10.

=

dv =

= ln x

= (en x)(en(en x)) –

= (en x)(en(en x)) – en x +C

Hal 56

1.

Misal u = x-2

du = dx

= 3

2.

Misal u =

du = 10x dx

xdx =

3.

du = dx

8.

8.

4.

du =

= en(enx) + C

5.

Misal u = enx

du =

- con u+C

- con (en x) + C

6.

x = 2u

arc sin

7.

u = 2w → du = 2dw

u2 = 4w

2

8.

Misal u = 2x2-1

du = 2.(2x) dx

= 4x dx

→ x dx =

9.

Misal u = sinx

Du = cosx dx

e4 +C

esinx

+ C

10.

Misal u = 4x

du = 4dx

dx =

11.

Misal u = en x

du =

12.

p =

misal

c = 5

p =

=

=

= 2

=

=

13.

a2 = 3

a =

=

14.

Misal g(x) = x2+2x+4

g1(x) = 2x +2

en (g(x))+C

= en

g(x) =x2+6x+13

g1(x) = 2x+6

x+5 = (2x+6)/2+2

= (2x+6)+2

x+5 = g1(x)+2

=

Misal u = x+3

du = dx

=

Soal Halaman 61

1.

g1(x) = 2x+10

x+5 =

2.

g1(x) = 3x

2+15+0

= 3(x2+15)

x2+5 =

=

=

3. dx

Misal g(x) = cos x

G1(x) = -sin x

4.

c = 7

p =

=

=

g(x) = x2+4x+7

g1(x) = 2x+4

1

= 3x+6

3x+2 = (3x+6)-4

= 1

1

2b = 4 → b = 2

c = 7

p =

1

5.

5x+1 = 2 +14

=

= 2

2b = 6 → b = 3 ; c = 13

p =

=

=

p = 2

2

6.

2b = -6

b = -3

c = 13

g(x) = x2-6x+13

g1(x) = 2x-6

2g1(x) =4x-12

4x+1 = 4x-12+13

2(g(x)) +13

2 en

7.

1

= 3x – 6 +4

= 3x -2

2b = - 4

b = -2

c = 7

p = 2

=

=

1

Hal 62.

1.

=

=

Misal u = x2H

du = 2x dx → xdx =

=

=

2.

6x+8 ← sisa

g(x) = x2+6x+13

g1(x) = 2x + 6

3g1(x) = 6x+18

2b = 6

b = 3

c = 13

p =

Hal 70.

2).

x + 3 =

x = 1

4 = 5C

C =

x = -4

-1 = A

-1 = A

A =

x = 0

3 = A -4B + 4C

3 =

3 = -4B +

4B =

B =

-

Hal 70

1)

x = 1

1 = 5B

x = -4

-4 = A(-5)

A =

3).

Misal = u = x2

+ 1

du = 2x dx

x dx =

=

= -

4).

(Ax+B)(x2-1)+(Cx+D)

X = (Ax+B)(x2-1)(x

2+1)+(Cx+D)(x

2+1)+(Ex+F)

x = 0

0 = -B + D + F

x = 1

1 = 2C + 2D

x = -1

=

4D = 0

Contoh 18 :

Bila x = 0

-1 = A (1)2 + (B0 + C) (0 – 2)(0 +1) + (D0 + E) (0 – 2)

-1 = A + (1 – 2) (+1) + E(-2)

-1 = A + (-2)C + (-2) E

-1 = A – 2C – 2E

= A-2C-2E = -1 …….(1)

Bila x = 2

3.23 - 2.2

2 + 5.2 – 1 = A(2

2 + 1)

2 + (B2 + C) (0)( 1) + (D2 + E) (0)

3.0 - 2.4 + 10 – 1 = A (5)2

24 – 8 + 10 – 1 = 25 A

25 = 25 A

A = 1 …… (2)

Bila x = 1

3.13 - 2.1

2 + 5.1 – 1 = A(1

2 + 1)

2 + (B1 + C) (-1)( 2) + (D1 + E) (-1)

3 - 2+ 5 – 1 = 4A + (-2)(B+C) + (-1) (D+E)

5 = 4A – 2B – 2C –D - E

4A – 2B – 2C –D - E = 5 ……. (3)

A = 1 …… (2)

Bila x = i (= )

3(i)3 – 2(i)

2 + 5i – 1 = A.0 + (Bi + C)(x-2)(0) + Di(Di + E)(i – 2)

3i (-1). 2(-2) + 5i -1 = (Di + E) (I – 2)

-3i + 2 + 5i – 1 = (i - 2) Di + (i-2) E

2i + 1 = Di2 – 2 Di + Ei – 2E

2i + 1 = -D – 2Di + Ei – 2E

2i + 1 = (-2 Di + Ei) – D – 2E

2i + 1 = (-2D + E) I + (-D – 2E)

(-2D + E) = 2 ………….. (4)

(4) -2D + E = 2

(5) X 2 -2D + 4E = 2 -

0 + 5E = 0

E = 0

-2D + E = 2

-2D + 0 = 2

D = -1

(1) A – 2C – 2E = -1

1 – 2C – 0 = -1

-2C = -2

C = 1

(3) 4A – 2B – 2C –D – E = 5

4.1 – 2B – 2.1 – (-1)-0 = 5

4 – 2B – 2 + 1 = 5

-2B – 2 = 0

-2B = 2

B = -1

Atau

Jadi

Misal u = x2 + 1

du = 2x dx → x dx =

=

Misal u = x2 + 1

du = 2x dx

x dx =

=

=

=

= en

Hal 72

5).

x2-x

2x4 – 2x

3 -

3x – 2

3x – 2 = A(x – 1) +Bx

Bila x = 0 → -2 = A(-1)+0

A = 2

Bila x = 1 → 3.1.2 = 0 + B.1

B = 1

= 2 en

=

Hal 70

6).

4x3+x

2+1 = (Ax + B)(x

2 - 2)

2 + (Cx + D)(x

2 – 2) + (Ex + F)

x = 0 → 1 = B(-2)2 + (D)(-2) + (F)

1 = 4B – 2D + F

4B – 2D + F = 1

x =

5 =

8

→ E = 8

F = 3

x = 1

6 =

6 =

A + B – C – D = 6 – 11

A + B – C – D = -5

x = -1

-2 =

-2 = -A + B + C – D - 5

-A + B + C – D = 3

x = 2

4.8 + 4 +1 =

32 + 5 =

8A + 4B + 4C + 2D = 37 – 19

8A + 4B + 4C + 2D = 18

4A + 2B + 2C + D = 9

(1). 4B – 2D + (3) = 1

4B – 2D = -2

2B – D = -1

(4). A + B – C – D = -5

(5). -A + B + C – D =3

2B – 2D = -2

B – D = -1

(1). 2B – D = -1

(7). B – D = -1

B = 0

(9). B – D = -1

0 – D = -1

D = 1

(4). A + B – C – D = -5

(5). -A + B + C – D = 3

2A – 2C = -2

A – C = -4

(6). 4A + 2B + 2C + D = 9

4A + 2(0) + 2C + (1) = 9

4A + 2C + 1 = 9

4A + 2C = 8

2A + C = 4

(8). A – C = -4

(9). 2A + C = 4

3A = 0

A = 0

(5). A – C = -4

0 – C = -4

C = 4

Misal u = x2 – 2

du = 2x dx

= 2 (-1)(u-1

)

=

=

Misal x =

u = x –

du = dx

g (x) = x2 – 2

g1(x) = 2x

u = x2 – 2

du = 2x dx

4

4

4

Hal 71

Sin x sin y =

Sin x cos y =

Cos x cos y =

Sin2x =

Cos2x =

1).

Sin 3x sin 2x =

=

=

2).

Sin 3x cosx =

=

3).

Cos 3x cos 2x =

=

4).

5).

= sin(2x)

Cos2(2x) =

=

= +

Hal 72

Contoh 22

1).

=

=

=

=

Hal 71

Sin x sin y =

Sin x cos y =

Cos x cos y =

Sin2x =

Cos2x =