first derivative from x^y = y^x

Find the first derivative from

x^y = y^x

Answare:

Use logarithmic derivative to find the first derivative this function

ln x^y = ln y^x

y ln x = x ln y

y = (x ln⁡y) / ln⁡x

so:

d/dx (x ln⁡ y) = d/dx (x) ln y + x d/dx (ln y)

= ln y + x [d/dy (ln y)  dy/dx]

= ln y + x ( 1/y) dy/dx

 = ln y +  x/y dy/dx

So:

y = (x ln⁡y) / ln ⁡x

dy/dx = [ d/dx(x lny) ln⁡ x – x ln⁡ y d/dx (ln⁡ x) ] / (ln⁡ x)^2

dy/dx = [ ( ln y + x/y dy/dx) ln⁡ x – x ln⁡ y 1/x] / (ln⁡ x)^2

dy/dx = [ ln⁡ y ln⁡ x + ln⁡x ( x/y)  dy/dx –  ln⁡ y ] / (ln⁡ x)^2

ln^2 x dy/dx =  ln⁡ y ln⁡ x + ln⁡x ( x/y) dy/dx –  ln⁡ y

ln^2 x dy/dx – x/y ln⁡ x dy/dx = ln⁡ y ln⁡ x – ln ⁡y

dy/dx (ln^2 x  –  x/y ln⁡x ) = ln⁡ y (ln⁡x – 1)

dy/dx = [ln⁡ y ln⁡ (x-1)] / [ln^2 x – x/y ln⁡ x]

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Find the first derivative from x^y = y^x

integral dari (a) ∫ sin⁡ 2x cos⁡ 3x dx (b) ∫ sin^2 3x cos⁡ 3x dx

Tentukanlah integral dari:

1. ∫ sin⁡ 2x cos⁡ 3x dx

2. ∫ sin^2 3x cos⁡ 3x dx

Jawab:

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

Maka:

∫ sin⁡ 2x cos⁡ 3x dx = -1 /2 [ cos⁡(2-3)x /(2-3) + (cos ( 2+3)x) /(2+3)] + C

∫ sin⁡ 2x cos⁡ 3x dx =  – 1 /2 [ cos⁡(-x) /(-1) + (cos 5x) /5 ] + C

∫ sin⁡ 2x cos⁡ 3x dx =  -cos ⁡5x /10 + cos⁡x /2  + C

2. Misalkan: u = sin 3x

Maka: du/dx = 3 cos⁡3x

Sehingga: dx = du /(3 cos⁡3x )

∫ sin^2 3x cos⁡ 3x dx =  ∫ u^2 cos⁡ 3x  (du/3 cos ⁡3x )

∫ sin^2 3x cos⁡ 3x dx = 1 /3 ∫u^2  du

∫ sin^2 3x cos⁡ 3x dx = 1/3 1/3 u^3 + C

∫ sin^2 3x cos⁡ 3x dx = 1/9 (sin 3x)^3 +  C

∫ sin^2 3x cos⁡ 3x dx = 1/9 sin^3 3x +  C

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Tentukanlah integral dari:

1. ∫ sin⁡ 2x cos⁡ 3x dx

2. ∫ sin^2 3x cos⁡ 3x dx

Integral dari ∫ sin^4 x cos^3 x dx

Tentukanlah Integral dari :

∫ sin^4 x cos^3 x dx

Jawab:

Misalkan: u = sin x

Maka: du/dx = cos⁡x

Sehingga: dx = du/cos⁡x

∫ sin^4 x cos^3 x dx = ∫u^4 cos^3 x du/cos⁡x

∫ sin^4 x cos^3 x dx = ∫ u^4 cos^3 x (du/cos⁡ x)

∫ sin^4 x cos^3 x dx = ∫ u^4 cos^2 x du

Karena: cos^2 x = 1 – sin^2 x

Maka:

∫ sin^4 x cos^3 x dx =  ∫ u^4 (1- sin^2 x) du

Karena: u = sin x

Maka:

∫ sin^4 x cos^3 x dx =  ∫ u^4 (1- u^2) du

∫ sin^4 x cos^3 x dx =  ∫ (u^4- u^6) du

∫ sin^4 x cos^3 x dx = 1 /(4+1) u^(4+1) – 1 /(6+1) u^(6+1) +  C

∫ sin^4 x cos^3 x dx = 1 /5 u^5 – 1 /7 u^7 +  C

∫ sin^4 x  cos^3 x dx =  1 /5 sin^5 x –  1 /7 sin^7 x +  C

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Tentukanlah Integral dari : ∫ sin〗^4 x cos^3 x dx

integral dari ∫ sin^4 x dx

Tentukanlah integral dari : ∫ sin^4 x dx

Jawab:

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

Maka:

∫ sin^4 x dx = – (sin^(4-1) x cos⁡x) /4 +  (4-1)/4 ∫ sin^(4-2) x dx

∫ sin^4 x dx = – (sin^3 x cos⁡x) /4 +  3/4 ∫ sin^2 x dx

∫ sin^4 x dx = – (sin^3 x cos⁡x) /4 + 3/4 [- (sin^(2-1) x cos⁡x) /2 + (2-1)/2 ∫ sin^(2-2) x dx ]

∫ sin^4 x dx = – (sin^3 x cos⁡x) /4  + 3/4 [- (sin^1 x cos⁡x) /2 + 1/2 ∫ sin^0 x dx ]

∫ sin^4 x dx = – (sin^3 x cos⁡x) /4 + 3/4 [- (sin x cos⁡x)/2 + 1/2 ∫ dx ]

∫ sin^4 x dx = – ( sin^3 x cos⁡x) /4 +  3/4 [- (sin x cos⁡x)/2+1/2 (x) ] + C

∫ sin^4 x dx =  – (sin^3 x cos⁡x) /4 – (3 sin x cos⁡x) / 8 +  (3 x) / 8 + C

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Tentukanlah integral dari : ∫ sin^4 x dx

integral dari : ∫ cos^4 x dx

Tentukanlah integral dari : ∫ cos^4 x dx

Jawab:

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

Maka:

∫ cos^4 x dx = (cos^3 x sin ⁡x) /4 + (4-1) /4 ∫ cos^2 x dx

∫ cos^4 x dx =  (cos^3 x sin ⁡x) /4 + 3/4 ( ( cos x sin ⁡x) /2 + (2-1)/2 ∫ (cos⁡x)^(2-2) dx )

∫ cos^4 x dx = (cos^3 x sin ⁡x) /4 + 3/4 ( ( cos x sin ⁡x) /2 + 1/2 ∫ (cos⁡x)^0 dx )

∫ cos^4 x dx = (cos^3 x sin ⁡x) /4 + 3/4 ( ( cos x sin ⁡x) /2 + 1/2 ∫ dx )

∫ cos^4 x dx = (cos^3 x sin ⁡x) /4 +  3/4 (( cos x sin ⁡x) /2 + 1/2 (x) )  + C

∫ cos^4 x dx =  (cos^3 x  sin ⁡x) /4 + ( 3 cos x sin ⁡x) /8 + 3/8 x + C

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Tentukanlah integral dari : ∫ cos^4 x dx