(a) ∫ (2x+10)dx (b)∫ √(8+5x) dx (c)∫ dx /(3x+1)^2

Hitunglah integral berikut:

1. ∫ (2x+10)dx

2. ∫ √(8+5x) dx

3. ∫ dx /(3x+1)^2

Jawab:

1.∫ (2x+10) dx = 2/(1+1) x^(1+1)+ 10x+C =  x^2+ 10x+C

2. Misalkan : u = 8 + 5x

Maka: du/dx=5 Sehingga: dx = 1/5 du

∫ √(8+5x) dx = ∫ (u)^(1/2) 1/5 du = 1/5 ∫ (u)^(1/2) du = (1/5) 1/(1/2+ 1) u^(1/2+1)+ C =  (1/5) 2/3 u^(3/2)+ C = 2/15 (8+5x)^(3/2)+ C= 2/15 (8+5x) √(8+5x)+C

3. Misalkan: u = 3x+1

Maka: du/dx=3 Sehingga : dx = du/3

∫ dx /(3x+1)^2 = ∫ (du/3) /(u)^2 = 1/3 ∫ du /(u)^2 =  1/3 1/(-2+1) (u)^(-2+1) +C = 1 /(-3) u^(-1) + C =  1 / (-3) (3x+1)^(-1) + C = (-1) /(9x+3) +  C

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Hitunglah integral berikut:

1. ∫ (2x+10)dx

2. ∫ √(8+5x) dx

3. ∫ dx /(3x+1)^2

Rumus lengkap Integral tak tentu

Rumus lengkap Integral tak tentu

1. ∫du = u+c

2. ∫a du =  a ∫du

3. ∫(du+dv) = ∫du+ ∫dv

4. ∫(du-dv) = ∫du- ∫dv

5. ∫u^n du = u^n/(n+1)+ C ,n tidak sama dengan-1

6. ∫du/u = ln⁡| u | + C

7. ∫e^u du =  e^u+ c

8. ∫a^u du =  a^u / ln⁡a + C      ,a>0 dan a ≠1

9. ∫√(a^2- x^2 ) dx =  x/2 √(a^2- x^2 )+ a^2/2 arc sin⁡ (x/a)+ C

Rumus Integral untuk fungsi Trigonometri

1. ∫sin⁡ u du = – cos u+C

2. ∫cos ⁡u du = sin⁡ u+C

3. ∫ sec^2⁡ u du = tg⁡ u+C

4. ∫ cosec^2 u du = – ctg u+C

5. ∫ sec⁡ u tg u du =  sec⁡ u+C

6. ∫ cosec⁡ u ctg u du = – cosec u+C

7. ∫ sin^n ax cos⁡ ax dx = 1/a ∫ u^n du =  1/a u^(n+1)/(n+1)+C   ,n ≠ -1 dan u=sin⁡ax

8. ∫ sin^n ax cos⁡ ax dx =  1/a ∫ u^n du =  1/a ln⁡| u |+C   ,n= -1 dan u = sinax

9. ∫ sin ^n u du =  – (sin^(n-1) u cos⁡u)/n+ (n-1)/n ∫ sin^(n-2) u du+c

10. ∫ cos ^n u du= (cos^(n-1) u sin ⁡u)/n+ (n-1)/n ∫ cos^(n-2) u du+c

11. ∫ sin⁡ ax sin⁡ bx dx = 1/2 [ sin⁡(a-b)x/(a-b)- (sin ( a+b)x)/(a+b)]+C

12. ∫ sin⁡ ax cos⁡ bx dx = -1/2 [ cos⁡(a-b)x/(a-b)+ (cos ( a+b)x)/(a+b)]+C

13. ∫ cos⁡ ax cos⁡ bx dx =  1/2 [ sin⁡(a-b)x/(a-b)+ (sin ( a+b)x)/(a+b)]+C

14. ∫ sin^2 ax dx =  1/2 (x-sin⁡2ax/2a) +C

15. ∫ cos^2 ax dx =  1/2 (x+ sin⁡2ax/2a) +C

16. ∫ x^n sin⁡ ax dx = (-1) / a x^n cos⁡ ax+  n/a ∫ x^(n-1) cos⁡ ax dx

17. ∫ x^n cos⁡ ax dx = 1 / a x^n sin⁡ ax+  n/a ∫ x^(n-1) sin⁡ ax dx

Rumus Integral dari Fungsi invers trigonometri

1. ∫ du/√(1- u^2 ) = arc sin⁡ u+C

2. ∫ -du/√(1- u^2 ) = arc  cos u+C

3. ∫ du/(1+u^2 ) = arc tg u+C

4. ∫ -du/(1+u^2 ) =  arc  ctg u+C

5. ∫ du/(u √( u^2- 1)) = arc sec |u|+C

6. ∫ -du/(u √( u^2- 1)) = arc  cosec |u|+C

Rumus integral Parsial

∫ v du= uv- ∫ u dv

Rumus Integral Rangkap dua melalui daerah tertutup s

1. ∬ f(x,y) dxdy = ∫_(b_1)^(b_2) ∫_(x_1)^(x_2) f(x,y) dxdy

2. ∬f(x,y) dxdy = ∫_(a_1)^(a_2) ∫_(y_1)^(y_2) f(x,y) dxdy

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Rumus lengkap Integral tak tentu Rumus Integral untuk fungsi Trigonometri Rumus Integral dari Fungsi invers trigonometri

Turunan pertama dari fungsi hiperbol trigonometri berikut! y = x sinh x dan y = x^2/cosh⁡ x

Tentukanlah Turunan pertama dari fungsi hiperbol trigonometri berikut!

1. y = x sinh x

2. y = x^2/cosh⁡ x

Jawab:

1. dy/dx=1(sinh⁡ x) – x d/dx(sinh⁡ x)

dy/dx = sinh⁡ x + x cosh ⁡x

Atau

dy/dx = (e^x- e^(-x) )/2⁡+ x ((e^x+ e^(-x) )/2)

2. dy/dx =  (2x(cosh⁡ x) –  x^2 (sinh⁡ x)) / (cosh x)^2

Atau :

dy/dx =  (2x((e^x+ e^(-x) )/2)-x^2 ((e^x- e^(-x) )/2) )/((e^x+ e^(-x) )/2)^2

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Tentukanlah Turunan pertama dari fungsi hiperbol  trigonometri berikut! 1.y = x sinh x 2.y = x^2/cosh⁡ x

Rumus Turunan Pertama dari fungsi Invers Trigonometri dan fungsi hiperbol Trigonometri

Rumus Turunan Pertama dari fungsi Invers Trigonometri

1. d/dx(arc sin⁡ u )= 1/√(1- u^2 ) (du/dx)

2. d/dx(arc cos⁡ u )= (-1)/√(1- u^2 ) (du/dx)

3. d/dx ( arc tg u)= 1/(1+ u^2 ) (du/dx)

4. d/dx ( arc sec u)= 1/(|u| √(u^2- 1)) (du/dx)

5. d/dx ( arc ctg u)= (-1)/(1+ u^2 ) (du/dx)

6. d/dx ( arc cosec u)= (-1)/(|u| √(u^2- 1)) (du/dx)

Fungsi Hiperbol Trogonometri

1. Sinh = (e^x- e^(-x) )/2 cosh = (e^x+ e^(-x) )/2

2. tanh = (e^x- e^(-x) )/(e^x+ e^(-x) )

3. coth = (e^x+ e^(-x) )/(e^x- e^(-x) )

4. sech = (2 )/(e^x+ e^(-x) )

1. d/dx(sinh⁡ u )  = -coshu (du/dx) =  (e^x+ e^(-x) )/2 (du/dx)

2. d/dx(cosh⁡ u ) =  sinhu (du/dx) =  (e^x- e^(-x) )/2 (du/dx)

3. d/dx(tanh⁡ u ) =  sech^2 u (du/dx) = ((2 )/(e^x+ e^(-x) ))^2 (du/dx)

4. d/dx(coth⁡ u ) =Rumus Turunan Pertama dari fungsi hiperbol Trigonometri  – cosech^2 u (du/dx) = -((2 )/(e^x- e^(-x) ))^2 (du/dx)

5. d/dx(sech⁡ u ) = – sech u tanh⁡ u (du/dx) =  – ((2 )/(e^x+ e^(-x) ))^2 ((e^x- e^(-x) )/(e^x+ e^(-x) ))(du/dx)

6. d/dx(cosech⁡ u ) = – cosech u coth⁡ u (du/dx) =  – ((2 )/(e^x- e^(-x) ))^2 ((e^x+ e^(-x) )/(e^x- e^(-x) ))(du/dx)

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Rumus Turunan Pertama dari fungsi Invers Trigonometri , Fungsi Hiperbol Trogonometri , Rumus Turunan Pertama dari fungsi hiperbol Trigonometri

y = arc tg (x – 1)/(x +1)

Tentukan turunan pertama dari :

1. y = arc tg (x – 1)/(x +1) !

Jawab:

Karena: d/dx ( arc tg u)= 1/(1+ u^2 ) (du/dx)

d/dx (arc tg (x – 1)/(x +1))= 1/(1+ ((x – 1)/(x +1))^2 ) (d((x – 1)/(x +1))/dx)

d/dx (arc tg (x – 1)/(x +1))= 1/(((x+1)^2+ (x-1)^2)/(x+1)^2 ) [(1(x+1)- (x-1)1)/(x+1)^2 ]

d/dx (arc tg (x – 1)/(x +1))= (x+1)^2/((x+1)^2+ (x-1)^2 ) (2/(x+1)^2 )

d/dx (arc tg (x – 1)/(x +1))= 2/((x+1)^2+ (x-1)^2 )

d/dx (arc tg (x – 1)/(x +1))= 2/(( x^2+ 2x+1)+ (x^2- 2x+1))

d/dx (arc tg (x – 1)/(x +1))= 2/(2x^2+ 2)

d/dx (arc tg (x – 1)/(x +1))= 2/(2(x^2+ 1))

d/dx (arc tg (x – 1)/(x +1))= 1/(x^2+ 1)

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Tentukan turunan pertama dari y = arc tg (x – 1)/(x +1)