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Kalkulus II Integral Fungsi Transenden Dr. Eko Pujiyanto, S.Si., M.T. www.ekopujiyanto.wordpress.com [email protected] 081 2278 3991

Materi Integral Fungsi Logaritma dan Eksponen Integra Invers Fungsi Trigonometri Integra Fungsi Hiperbolik

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Materi ke - 3

Integral Fungsi Logaritma dan Eksponen Teorema 1

dx (a ) ∫ = ln x + C x

(b ) ∫ e

x

dx = e + C x

Rumus Integral Parsial

udv = uv − vdu ∫ ∫ [email protected]

[email protected] kalkulus 2 - materi ke 2

Materi ke - 3

Integral Fungsi Logaritma dan Eksponen Contoh 1

( xdx x −1 +1 x − 1) + 1 ∫ x − 1 = ∫ x − 1 dx = ∫ x − 1 dx ( 1 x − 1) =∫ dx + ∫ dx x −1 x −1 1 = ∫ 1.dx + ∫ dx = x + ln x − 1 + C x −1 [email protected]


Materi ke - 3


1+ ex 1 − e x + 2e x ∫ 1 − e x dx = ∫ 1 − e x dx 1− ex 2e x =∫ dx + ∫ dx x x 1− e 1− e x 2e x = ∫ 1.dx + ∫ dx = x − 2 ln 1 − e + C x 1− e [email protected]


Materi ke - 3


∫ ln x dx ⇒ u = ln x du 1 = dx x

& dv = dx & v=x

ln x dx = x ln x − dx = x ln x − x + C ∫ ∫

[email protected]


Materi ke - 3

Integral Fungsi Logaritma dan Eksponen Teorema 2 x

a ∫ a dx = ln a + C , a > 0 & a ≠ 1 x

[email protected]


Materi ke - 3


2 − ln x dx ∫1 x dx ⇒ u = ln x → du = x e

−u −u 2 du = − 2 ∫ ∫ d (−u ) −u

− ln x

 2 2 =− →  − ln 2  ln 2

e

(

 1 −ln e  = − 2 − 2 −ln1 ln 2 1 1 −1 −0 1 −1 =− 2 −2 = − 2 −1 ln 2 ln 2 1  1 1 1 =− = 0.5 * 0.69 = 0.3 −  = ln 2  2  2 ln 2

(

[email protected]

)


(

)

Materi ke - 3

)

Integral Fungsi Logaritma dan Eksponen Latihan

1− e 1. ∫ dx x 1+ e x

2. ∫ e

sin x

sin 2 xdx

3. ∫ ln xdx 2

4.∫ sin (ln x )dx [email protected]


Materi ke - 3

Integral Fungsi Logaritma dan Eksponen Latihan

[email protected]


Materi ke - 3

Integral Fungsi Trigonometri dan Inversnya Contoh 5

sin x (a ) ∫ tan xdx = ∫ dx = − ln cos x + C cos x cos x (b ) ∫ cot xdx = ∫ dx = ln sin x + C sin x

[email protected]


Materi ke - 3

Integral Fungsi Trigonometri dan Inversnya Teorema 3 x (a ) ∫ = sin x + C ⇒ ∫ = sin +C 2 2 a 1− x a−x dx dx 1 −1 −1 x (b ) ∫ = tan x + C ⇒ ∫ = tan +C 2 2 1+ x a+x a a dx dx 1 −1 −1 x (c ) ∫ = sec x + C ⇒ ∫ = sec +C a x 1− x2 x a − x2 a dx

[email protected]

−1

dx


−1

Materi ke - 3


x +1 1 2x + 2 ∫ x 2 − 4 x + 5dx = 2 ∫ x 2 − 4 x + 5 dx 1 2x − 4 + 6 = ∫ 2 dx 2 x − 4x + 5 1 2x − 4 1 6 = ∫ 2 dx + ∫ 2 dx 2 x − 4x + 5 2 x − 4x + 5 A [email protected]


B Materi ke - 3

Integral Fungsi Trigonometri dan Inversnya A 1 2x − 4 = ∫ 2 dx 2 x − 4x + 5 u = x2 − 4x + 5 du = (2 x − 4 )dx

B 1 6 dx = ∫ 2 2 x − 4x + 5 1 = 3∫ dx 2 1 + ( x − 2)

= 3 tan −1 ( x − 2 ) + C

1 du 1 = ∫ = ln u 2 u 2 1 2 = ln x − 4 x + 5 + C 2 [email protected] [email protected]

kalkulus 2 - materi ke 2

Materi ke - 3


∫

1 − 2 x − 2 + 2 dx =− ∫ 2 2 2 − 2x − x − 2x − x ( − 2 x − 2 ) dx 1 2 dx =∫ − ∫ 2 2 2 − 2x − x − 2x − x x dx

⇓ [email protected]

⇓ [email protected] kalkulus 2 - materi ke 2

Materi ke - 3


⇓

⇓

u = −2 x − x

dx

=∫

2

du = (− 2 − 2 x )dx

1 − ( x − 1) 2 −1

= sin ( x − 1) + C

⇓ 1 du =− ∫ = − u + C = − − 2x − x2 + C 2 u [email protected]


Materi ke - 3


(

)

ln 1 + x dx ∫ 2

(

u = ln 1 + x

2

)

& dv = dx

2 x dx du = & v=x 2 1+ x

(

)

(

)

2

x dx ∫ ln 1 + x dx = x ln 1 + x − 2∫ 1 + x 2 [email protected]

2

2


Materi ke - 3


(

)

(

)

2 x dx 2 2 ∫ ln 1 + x dx = x ln 1 + x − 2∫ 1 + x 2 2 1 + x − 1 dx 2 = x ln 1 + x − 2 ∫ 2 1+ x dx 2 = x ln 1 + x − 2 ∫ dx + 2 ∫ 1+ x2 2 −1 = x ln [email protected] + x − 2 x − 2 tan x + C [email protected]

(

)

( (

) )


(

)

Materi ke - 3

Integral Fungsi Trigonometri dan Inversnya Latihan

−1

1. ∫ e tan e dx x

x

−1

2. ∫ x sin xdx 3. ∫

dx e +1 x

(

)

4.∫ ln 1 − x dx [email protected]

2


Materi ke - 3

Integral Fungsi Trigonometri dan Inversnya Latihan

[email protected]


Materi ke - 3

Integral Fungsi Hiperbolik dan Inversnya Teorema 3

1. ∫ sinh x dx = cosh x + C 2. ∫ cosh x dx = sinh x + C . . 8. ∫ coth x dx = ln sinh x + C [email protected]


Materi ke - 3

Integral Fungsi Hiperbolik dan Inversnya Contoh 9 x

e dx 2dx 1. ∫ sech x dx = ∫ x − x = 2 ∫ 2 x e +e e +1 −1 x = 2 tan e + C dx cosh dx 2. ∫ sech x dx = ∫ =∫ cosh x cosh 2 x cosh xdx −1 =∫ = tan sinh x + C 2 1 + sinh x [email protected] [email protected]


Materi ke - 3

Integral Fungsi Hiperbolik dan Inversnya Teorema 4 1. ∫ 2. ∫

dx x +1 dx 2

x2 −1

( x = ln (x +

) − 1 )+ C

= sinh −1 x = ln x + x 2 + 1 + C = cosh −1

x2

1 1+ x  −1 tanh x + C = ln +C , x <1  dx  2 1− x 3. ∫ = 2 1− x coth −1 x + C = 1 ln 1 + x + C , x > 1  2 1− x [email protected]


Materi ke - 3

Integral Fungsi Hiperbolik dan Inversnya Contoh 10 dx

∫

x − 4x 2

=∫ =∫

dx

(x − 2)

2

− 22

dx

 x−2   −1  2  −1 x − 2 = cosh +C 2 [email protected] [email protected]

2


Materi ke - 3

Integral Fungsi Hiperbolik dan Inversnya Latihan 1.

∫x

2

cosh xdx

sinh 2 xdx 2. ∫ 4 1 + cosh x 3. ∫ sinh xdx 3

4. ∫ [email protected]

dx x − 9x 2


Materi ke - 3

Integral Fungsi Hiperbolik dan Inversnya Latihan

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Materi ke - 3

Kata Inspirasi Hari Ini Pendengar yang baik akan mampu menerima informasi dari banyak orang.

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Materi ke - 3

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