integral de (3x)/(x^2+10x+30)

Solución

\int \frac{3 x}{x ^{2} + 10 x + 30} d x

3 (\frac{1}{2} ln \frac{1}{5} (x + 5)^{2} + 1 - 5 arctan (\frac{1}{5} (x + 5))) + C

Pasos de solución

\int \frac{3 x}{x ^{2} + 10 x + 30} d x

Sacar la constante:

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

= 3 \cdot \int \frac{x}{x ^{2} + 10 x + 30} d x

Completar el cuadrado

x^{2} + 10 x + 30 : (x + 5)^{2} + 5

= 3 \cdot \int \frac{x}{( x + 5 ) ^{2} + 5} d x

Aplicar integración por sustitución

: \int \frac{u - 5}{u ^{2} + 5} d u

= 3 \cdot \int \frac{u - 5}{u ^{2} + 5} d u

Aplicar integración por sustitución

= 3 \cdot \int \frac{v - 5}{v ^{2} + 1} d v

Expandir

\frac{v - 5}{v ^{2} + 1} : \frac{v}{v ^{2} + 1} - \frac{5}{v ^{2} + 1}

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

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

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

\int \frac{5}{v ^{2} + 1} d v = 5 arctan (v)

= 3 (\frac{1}{2} ln v^{2} + 1 - 5 arctan (v))

Sustitución hacia atrás

= 3 (\frac{1}{2} ln (\frac{x + 5}{5})^{2} + 1 - 5 arctan (\frac{x + 5}{5}))

Simplificar

3 (\frac{1}{2} ln (\frac{x + 5}{5})^{2} + 1 - 5 arctan (\frac{x + 5}{5})) : 3 (\frac{1}{2} ln \frac{1}{5} (x + 5)^{2} + 1 - 5 arctan (\frac{1}{5} (x + 5)))

= 3 (\frac{1}{2} ln \frac{1}{5} (x + 5)^{2} + 1 - 5 arctan (\frac{1}{5} (x + 5)))

Agregar una constante a la solución

= 3 (\frac{1}{2} ln \frac{1}{5} (x + 5)^{2} + 1 - 5 arctan (\frac{1}{5} (x + 5))) + C

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