es

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral de (2x)/(9x^2-12x+8)

Solución

\int \frac{2 x}{9 x ^{2} - 12 x + 8} d x

Solución

\frac{1}{9} (ln 9 x^{2} - 12 x + 8 + 2 arctan (\frac{3 x - 2}{2})) + C

Pasos de solución

\int \frac{2 x}{9 x ^{2} - 12 x + 8} d x

Sacar la constante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{x}{9 x ^{2} - 12 x + 8} d x

Completar el cuadrado

9 x^{2} - 12 x + 8 : 9 (x - \frac{2}{3})^{2} + 4

= 2 \cdot \int \frac{x}{9 ( x - \frac{2}{3} ) ^{2} + 4} d x

Aplicar integración por sustitución

: \int \frac{3 u + 2}{3 ( 9 u ^{2} + 4 )} d u

= 2 \cdot \int \frac{3 u + 2}{3 ( 9 u ^{2} + 4 )} d u

Sacar la constante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{3} \cdot \int \frac{3 u + 2}{9 u ^{2} + 4} d u

Expandir

\frac{3 u + 2}{9 u ^{2} + 4} : \frac{3 u}{9 u ^{2} + 4} + \frac{2}{9 u ^{2} + 4}

Aplicar la regla de la suma:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 \cdot \frac{1}{3} (\int \frac{3 u}{9 u ^{2} + 4} d u + \int \frac{2}{9 u ^{2} + 4} d u)

\int \frac{3 u}{9 u ^{2} + 4} d u = \frac{1}{6} ln 9 u^{2} + 4

\int \frac{2}{9 u ^{2} + 4} d u = \frac{1}{3} arctan (\frac{3}{2} u)

= 2 \cdot \frac{1}{3} (\frac{1}{6} ln 9 u^{2} + 4 + \frac{1}{3} arctan (\frac{3}{2} u))

Sustituir en la ecuación

u = x - \frac{2}{3}

= 2 \cdot \frac{1}{3} (\frac{1}{6} ln 9 (x - \frac{2}{3})^{2} + 4 + \frac{1}{3} arctan (\frac{3}{2} (x - \frac{2}{3})))

Simplificar

2 \cdot \frac{1}{3} (\frac{1}{6} ln 9 (x - \frac{2}{3})^{2} + 4 + \frac{1}{3} arctan (\frac{3}{2} (x - \frac{2}{3}))) : \frac{1}{9} (ln 9 x^{2} - 12 x + 8 + 2 arctan (\frac{3 x - 2}{2}))

= \frac{1}{9} (ln 9 x^{2} - 12 x + 8 + 2 arctan (\frac{3 x - 2}{2}))

Agregar una constante a la solución

= \frac{1}{9} (ln 9 x^{2} - 12 x + 8 + 2 arctan (\frac{3 x - 2}{2})) + C

Ejemplos populares

tangent y=x+sqrt(x),(9,12)tangente de

y = x + x, (9, 12)

derivada de (x^3/(12)+2/3 x^{-3})

\frac{d}{d x} (\frac{x ^{3}}{12} + \frac{2}{3} x^{- 3})

derivada de-8e^{-x/(10}-40)

\frac{d}{d x} (- 8 e^{- \frac{x}{10}} - 40)

integral de 0 a 8 de 4/((x-6)^3)

\int_{0}^{8} \frac{4}{( x - 6 ) ^{3}} d x

tangent f(x)=(1+2x)^{10},(0,1)tangente de

f (x) = (1 + 2 x)^{10}, (0, 1)