integral de-2 a 0 de 2/(x^2+4x+8)

Solución

\int_{- 2}^{0} \frac{2}{x ^{2} + 4 x + 8} d x

\frac{π}{4}

Notación decimal

0.78539 \dots

Pasos de solución

\int_{- 2}^{0} \frac{2}{x ^{2} + 4 x + 8} d x

Sacar la constante:

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

= 2 \cdot \int_{- 2}^{0} \frac{1}{x ^{2} + 4 x + 8} d x

Completar el cuadrado

x^{2} + 4 x + 8 : (x + 2)^{2} + 4

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

Aplicar integración por sustitución

: \int_{0}^{2} \frac{1}{u ^{2} + 4} d u

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

Aplicar integración por sustitución

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

Sacar la constante:

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

= 2 \cdot \frac{1}{2} \cdot \int_{0}^{1} \frac{1}{v ^{2} + 1} d v

Aplicar la regla de integración:

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

= 2 \cdot \frac{1}{2} [arctan (v)]_{0}^{1}

Simplificar

2 \cdot \frac{1}{2} [arctan (v)]_{0}^{1} : [arctan (v)]_{0}^{1}

= [arctan (v)]_{0}^{1}

Calcular los limites:

\frac{π}{4}

= \frac{π}{4}

\int_{- 1}^{2} x + 2 x^{2} d x

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