English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral de (sin(x))/(cos^2(x)+cos(x)-6)

Solución

\int \frac{sin ( x )}{cos ^{2} ( x ) + cos ( x ) - 6} d x

- 2 (\frac{1}{10} ln 3 tan^{2} (\frac{x}{2}) + 1 - \frac{1}{10} ln tan^{2} (\frac{x}{2}) + 2) + C

Pasos de solución

\int \frac{sin ( x )}{cos ^{2} ( x ) + cos ( x ) - 6} d x

Re-escribir usando identidades trigonométricas

= \int \frac{2 sin ( x )}{cos ( 2 x ) + 2 cos ( x ) - 11} d x

Sacar la constante:

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

= 2 \cdot \int \frac{sin ( x )}{cos ( 2 x ) + 2 cos ( x ) - 11} d x

Re-escribir usando identidades trigonométricas

= 2 \cdot \int \frac{sin ( x )}{2 cos ^{2} ( x ) + 2 cos ( x ) - 12} d x

Aplicar integración por sustitución

: \int - \frac{u}{3 u ^{4} + 7 u ^{2} + 2} d u

= 2 \cdot \int - \frac{u}{3 u ^{4} + 7 u ^{2} + 2} d u

Sacar la constante:

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

= 2 (- \int \frac{u}{3 u ^{4} + 7 u ^{2} + 2} d u)

Tomar la fracción parcial de

\frac{u}{3 u ^{4} + 7 u ^{2} + 2} : \frac{3 u}{5 ( 3 u ^{2} + 1 )} - \frac{u}{5 ( u ^{2} + 2 )}

= 2 (- \int \frac{3 u}{5 ( 3 u ^{2} + 1 )} - \frac{u}{5 ( u ^{2} + 2 )} 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 (- (\int \frac{3 u}{5 ( 3 u ^{2} + 1 )} d u - \int \frac{u}{5 ( u ^{2} + 2 )} d u))

\int \frac{3 u}{5 ( 3 u ^{2} + 1 )} d u = \frac{1}{10} ln 3 u^{2} + 1

\int \frac{u}{5 ( u ^{2} + 2 )} d u = \frac{1}{10} ln u^{2} + 2

= 2 (- (\frac{1}{10} ln 3 u^{2} + 1 - \frac{1}{10} ln u^{2} + 2))

Sustituir en la ecuación

u = tan (\frac{x}{2})

= 2 (- (\frac{1}{10} ln 3 tan^{2} (\frac{x}{2}) + 1 - \frac{1}{10} ln tan^{2} (\frac{x}{2}) + 2))

Simplificar

= - 2 (\frac{1}{10} ln 3 tan^{2} (\frac{x}{2}) + 1 - \frac{1}{10} ln tan^{2} (\frac{x}{2}) + 2)

Agregar una constante a la solución

= - 2 (\frac{1}{10} ln 3 tan^{2} (\frac{x}{2}) + 1 - \frac{1}{10} ln tan^{2} (\frac{x}{2}) + 2) + C

f (x) = x^{2} - 3 x + 5, at x = 1

n = 0 \sum \infty n^{3}

f (x) = - 7 x^{3} - 8 x^{2} - 5 x, at x = - 3

\int \frac{sin ( 2 x )}{1 + cos ( 2 x )} d x

\int \frac{1}{( 7 x - 1 ) 49 x ^{2} - 14 x} d x