es

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral (2x)/(sqrt(9x^2-6x-8))

Solución

integral

\frac{2 x}{9 x ^{2} - 6 x - 8}

Solución

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

Pasos de solución

\int \frac{2 x}{9 x ^{2} - 6 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} - 6 x - 8} d x

Completar el cuadrado

9 x^{2} - 6 x - 8 : 9 (x - \frac{1}{3})^{2} - 9

= 2 \cdot \int \frac{x}{9 ( x - \frac{1}{3} ) ^{2} - 9} d x

Aplicar integración por sustitución

: \int \frac{3 u + 1}{9 u ^{2} - 1} d u

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

Sacar la constante:

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

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

Expandir

\frac{3 u + 1}{u ^{2} - 1} : \frac{3 u}{u ^{2} - 1} + \frac{1}{u ^{2} - 1}

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

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}{9} (\int \frac{3 u}{u ^{2} - 1} d u + \int \frac{1}{u ^{2} - 1} d u)

\int \frac{3 u}{u ^{2} - 1} d u = 3 u^{2} - 1

\int \frac{1}{u ^{2} - 1} d u = ln u^{2} - 1 + u

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

Sustituir en la ecuación

u = x - \frac{1}{3}

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

Simplificar

2 \cdot \frac{1}{9} 3 (x - \frac{1}{3})^{2} - 1 + ln (x - \frac{1}{3})^{2} - 1 + x - \frac{1}{3} : \frac{2}{9} 3 (x - \frac{1}{3})^{2} - 1 + ln (x - \frac{1}{3})^{2} - 1 + x - \frac{1}{3}

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

Agregar una constante a la solución

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

Ejemplos populares

integral de 2+cos^2(x)sin(x)

\int 2 + cos^{2} (x) sin (x) d x

integral de x/2 sin(x^2)

\int \frac{x}{2} sin (x^{2}) d x

integral de (x^3-x)/(x^2+4x+9)

\int \frac{x ^{3} - x}{x ^{2} + 4 x + 9} d x

integral de 1/((x-sqrt(x)))

\int \frac{1}{( x - x )} d x

integral de (3-5x+7x^2)/(3x^3-19x^2-14x)

\int \frac{3 - 5 x + 7 x ^{2}}{3 x ^{3} - 19 x ^{2} - 14 x} d x