Hasil dari integral 4x – 3 / akar pangkat 3 (6x – 4x^2)^2 dx adalah -\frac{3}{2} \sqrt[3]{6x - 4x^{2}}−

2

3

3

6x−4x

2

+ C. Hasil tersebut diperoleh dengan menggunakan integral substitusi. Integral adalah anti turunan atau lawan dari turunan. Bentuk umum integral tak tentu adalah ∫ f’(x) dx = f(x) + C. Rumus dasar Integral:

∫ axⁿ dx = \frac{a}{n+1} x^{n + 1}

n+1

a

x

n+1

+ C, dengan n ≠ –1

Pembahasan

\int {\frac{4x - 3}{\sqrt[3]{(6x - 4x^{2})^{2}}}} \: dx∫

3

(6x−4x

2

)

2

4x−3

dx

= \int {\frac{4x - 3}{(6x - 4x^{2})^{\frac{2}{3}}} \: dx

= \int {(4x - 3)(6x - 4x^{2})^{-\frac{2}{3}}} \: dx∫(4x−3)(6x−4x

2

)

−

3

2

dx

Misal

u = 6x – 4x²

du = 6 – 8x dx

du = –2(–3 + 4x) dx

du = –2(4x – 3) dx

dx = \frac{du}{-2(4x - 3)}

−2(4x−3)

du

= \int {(4x - 3)(6x - 4x^{2})^{-\frac{2}{3}}} \: dx∫(4x−3)(6x−4x

2

)

−

3

2

dx

= \int {(4x - 3)(u)^{-\frac{2}{3}}} \: \frac{du}{-2(4x - 3)}∫(4x−3)(u)

−

3

2

−2(4x−3)

du

= \int {\frac{1}{-2} (u)^{-\frac{2}{3}}} \: du∫

−2

1

(u)

−

3

2

du

= \frac{\frac{1}{-2}}{\frac{1}{3}} (u)^{\frac{1}{3}}

3

1

−2

1

(u)

3

1

+ C

= -\frac{3}{2} \sqrt[3]{u}−

2

3

3

u

+ C

= -\frac{3}{2} \sqrt[3]{6x - 4x^{2}}−

2

3

3

6x−4x

2

+ C

Cara lain

\int {f(x) \: . \: g^{n} (x)} \: dx = \frac{f(x) \: . \: g^{n + 1} (x)}{(n + 1) \: . \: g' (x)} + C∫f(x).g

n

(x)dx=

(n+1).g

′

(x)

f(x).g

n+1

(x)

+C

Jadi

\int {\frac{4x - 3}{\sqrt[3]{(6x - 4x^{2})^{2}}}} \: dx∫

3

(6x−4x

2

)

2

4x−3

dx

= \int {\frac{4x - 3}{(6x - 4x^{2})^{\frac{2}{3}}} \: dx

= \int {(4x - 3)(6x - 4x^{2})^{-\frac{2}{3}}} \: dx∫(4x−3)(6x−4x

2

)

−

3

2

dx

= \frac{(4x - 3)(6x - 4x^{2})^{-\frac{2}{3} + 1}}{(-\frac{2}{3} + 1) (6 - 8x)}

(−

3

2

+1)(6−8x)

(4x−3)(6x−4x

2

)

−

3

2

+1

+ C

= \frac{(4x - 3)(6x - 4x^{2})^{\frac{1}{3}}}{(\frac{1}{3}).-2(4x - 3)}

(

3

1

).−2(4x−3)

(4x−3)(6x−4x

2

)

3

1

+ C

= \frac{(6x - 4x^{2})^{\frac{1}{3}}}{-\frac{2}{3}}

−

3

2

(6x−4x

2

)

3

1

+ C

= -\frac{3}{2} \sqrt[3]{6x - 4x^{2}}−

2

3

3

6x−4x

2

+ C

gitu kalo ga salah