integral 1/((x^2+1)^2)

Solución

integral

\frac{1}{( x ^{2} + 1 ) ^{2}}

\frac{1}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) + C

Pasos de solución

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

Aplicar integración por sustitución trigonométrica

: \int \frac{1}{sec ^{4} ( u ) cos ^{2} ( u )} d u

= \int \frac{1}{sec ^{4} ( u ) cos ^{2} ( u )} d u

Re-escribir usando identidades trigonométricas

= \int \frac{1 + cos ( 2 u )}{2} d u

Sacar la constante:

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

= \frac{1}{2} \cdot \int 1 + cos (2 u) 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

= \frac{1}{2} (\int 1 d u + \int cos (2 u) d u)

\int 1 d u = u

\int cos (2 u) d u = \frac{1}{2} sin (2 u)

= \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Sustituir en la ecuación

u = arctan (x)

= \frac{1}{2} (arctan (x) + \frac{1}{2} sin (2 arctan (x)))

Simplificar

\frac{1}{2} (arctan (x) + \frac{1}{2} sin (2 arctan (x))) : \frac{1}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

= \frac{1}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

Agregar una constante a la solución

= \frac{1}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) + C

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