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integral de 0 a 1 de 1/((x^2+2)^2)

Solución

\int_{0}^{1} \frac{1}{( x ^{2} + 2 ) ^{2}} d x

Solución

\frac{2 arctan ( \frac{1}{2} ) + sin ( 2 arctan ( \frac{1}{2} ) )}{8 2}

+1

Notación decimal

0.19213 \dots

Pasos de solución

\int_{0}^{1} \frac{1}{( x ^{2} + 2 ) ^{2}} d x

Aplicar integración por sustitución

= \int_{0}^{\frac{1}{2}} \frac{2}{( 2 u ^{2} + 2 ) ^{2}} d u

Sacar la constante:

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

= 2 \cdot \int_{0}^{\frac{1}{2}} \frac{1}{( 2 u ^{2} + 2 ) ^{2}} d u

Factorizar

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

= 2 \cdot \int_{0}^{\frac{1}{2}} \frac{1}{4 ( u ^{2} + 1 ) ^{2}} d u

Sacar la constante:

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

= 2 \frac{1}{4} \cdot \int_{0}^{\frac{1}{2}} \frac{1}{( u ^{2} + 1 ) ^{2}} d u

Aplicar integración por sustitución trigonométrica

: \int_{0}^{a r c t a n (\frac{1}{2})} \frac{1}{sec ^{4} ( v ) cos ^{2} ( v )} d v

= 2 \frac{1}{4} \cdot \int_{0}^{a r c t a n (\frac{1}{2})} \frac{1}{sec ^{4} ( v ) cos ^{2} ( v )} d v

Re-escribir usando identidades trigonométricas

= 2 \frac{1}{4} \cdot \int_{0}^{a r c t a n (\frac{1}{2})} \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

= 2 \frac{1}{4} \cdot \frac{1}{2} \cdot \int_{0}^{a r c t a n (\frac{1}{2})} 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

= 2 \frac{1}{4} \cdot \frac{1}{2} (\int_{0}^{a r c t a n (\frac{1}{2})} 1 d v + \int_{0}^{a r c t a n (\frac{1}{2})} cos (2 v) d v)

\int_{0}^{a r c t a n (\frac{1}{2})} 1 d v = arctan (\frac{1}{2})

\int_{0}^{a r c t a n (\frac{1}{2})} cos (2 v) d v = \frac{sin ( 2 arctan ( \frac{1}{2} ) )}{2}

= 2 \frac{1}{4} \cdot \frac{1}{2} arctan (\frac{1}{2}) + \frac{sin ( 2 arctan ( \frac{1}{2} ) )}{2}

Simplificar

2 \frac{1}{4} \cdot \frac{1}{2} arctan (\frac{1}{2}) + \frac{sin ( 2 arctan ( \frac{1}{2} ) )}{2} : \frac{2 arctan ( \frac{1}{2} ) + sin ( 2 arctan ( \frac{1}{2} ) )}{8 2}

= \frac{2 arctan ( \frac{1}{2} ) + sin ( 2 arctan ( \frac{1}{2} ) )}{8 2}

Ejemplos populares

integral de 0 a 1 de (1-y^2)

\int_{0}^{1} (1 - y^{2}) d y

integral de 0 a 4 de 8x^2

\int_{0}^{4} 8 x^{2} d x

integral de 1 a 7 de (t^2-2t-24)

\int_{1}^{7} (t^{2} - 2 t - 24) d t

integral de 0 a 4 de x^2+2x-3

\int_{0}^{4} x^{2} + 2 x - 3 d x

integral de-8 a 4 de (-x^2-3x)

\int_{- 8}^{4} (- x^{2} - 3 x) d x