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integral de 1/(x^3sqrt(x^2+1))

Solución

\int \frac{1}{x ^{3} x ^{2} + 1} d x

\frac{1}{8} (tan^{2} (\frac{arctan ( x )}{2}) - 4 ln tan (\frac{arctan ( x )}{2}) - cot^{2} (\frac{arctan ( x )}{2})) + C

Pasos de solución

\int \frac{1}{x ^{3} x ^{2} + 1} d x

Aplicar integración por sustitución trigonométrica

: \int cot^{3} (u) sec (u) d u

= \int cot^{3} (u) sec (u) d u

Re-escribir usando identidades trigonométricas

= \int \frac{cos ^{2} ( u )}{sin ^{3} ( u )} d u

Aplicar integración por sustitución

: \int \frac{( - v ^{2} + 1 ) ^{2}}{4 v ^{3}} d v

= \int \frac{( - v ^{2} + 1 ) ^{2}}{4 v ^{3}} d v

Sacar la constante:

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

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

Expandir

\frac{( - v ^{2} + 1 ) ^{2}}{v ^{3}} : v - \frac{2}{v} + \frac{1}{v ^{3}}

= \frac{1}{4} \cdot \int v - \frac{2}{v} + \frac{1}{v ^{3}} d v

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

= \frac{1}{4} (\int v d v - \int \frac{2}{v} d v + \int \frac{1}{v ^{3}} d v)

\int v d v = \frac{v ^{2}}{2}

\int \frac{2}{v} d v = 2 ln ∣ v ∣

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

= \frac{1}{4} (\frac{v ^{2}}{2} - 2 ln ∣ v ∣ - \frac{1}{2 v ^{2}})

Sustitución hacia atrás

= \frac{1}{4} \frac{tan ^{2} ( \frac{a r c t a n ( x )}{2} )}{2} - 2 ln tan (\frac{arctan ( x )}{2}) - \frac{1}{2 tan ^{2} ( \frac{a r c t a n ( x )}{2} )}

Simplificar

\frac{1}{4} \frac{tan ^{2} ( \frac{a r c t a n ( x )}{2} )}{2} - 2 ln tan (\frac{arctan ( x )}{2}) - \frac{1}{2 tan ^{2} ( \frac{a r c t a n ( x )}{2} )} : \frac{1}{8} (tan^{2} (\frac{arctan ( x )}{2}) - 4 ln tan (\frac{arctan ( x )}{2}) - cot^{2} (\frac{arctan ( x )}{2}))

= \frac{1}{8} (tan^{2} (\frac{arctan ( x )}{2}) - 4 ln tan (\frac{arctan ( x )}{2}) - cot^{2} (\frac{arctan ( x )}{2}))

Agregar una constante a la solución

= \frac{1}{8} (tan^{2} (\frac{arctan ( x )}{2}) - 4 ln tan (\frac{arctan ( x )}{2}) - cot^{2} (\frac{arctan ( x )}{2})) + C

\int \frac{3 x ^{3} + 1}{x ^{3} - x ^{2} + x - 1} d x

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

\int \frac{1}{16 x ^{2} + 1} d x

\int (5 x - 3) d x

\int \frac{2 x ^{2} - 8 x - 8}{( x - 2 ) ( x ^{2} + 4 )} d x