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integral (22)/((121x^2+1)^2)

Solución

integral

\frac{22}{( 121 x ^{2} + 1 ) ^{2}}

arctan (11 x) + \frac{1}{2} sin (2 arctan (11 x)) + C

Pasos de solución

\int \frac{22}{( 121 x ^{2} + 1 ) ^{2}} d x

Sacar la constante:

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

= 22 \cdot \int \frac{1}{( 121 x ^{2} + 1 ) ^{2}} d x

Aplicar integración por sustitución

= 22 \cdot \int \frac{1}{11 ( u ^{2} + 1 ) ^{2}} d u

Sacar la constante:

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

= 22 \cdot \frac{1}{11} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u

Aplicar integración por sustitución trigonométrica

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

= 22 \cdot \frac{1}{11} \cdot \int \frac{1}{sec ^{4} ( v ) cos ^{2} ( v )} d v

Re-escribir usando identidades trigonométricas

= 22 \cdot \frac{1}{11} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

Sacar la constante:

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

= 22 \cdot \frac{1}{11} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) 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

= 22 \cdot \frac{1}{11} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

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

= 22 \cdot \frac{1}{11} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Sustitución hacia atrás

= 22 \cdot \frac{1}{11} \cdot \frac{1}{2} (arctan (11 x) + \frac{1}{2} sin (2 arctan (11 x)))

Simplificar

22 \cdot \frac{1}{11} \cdot \frac{1}{2} (arctan (11 x) + \frac{1}{2} sin (2 arctan (11 x))) : arctan (11 x) + \frac{1}{2} sin (2 arctan (11 x))

= arctan (11 x) + \frac{1}{2} sin (2 arctan (11 x))

Agregar una constante a la solución

= arctan (11 x) + \frac{1}{2} sin (2 arctan (11 x)) + C

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