integral de 0 a 6 de sqrt(5x^2+10x+6)

Solución

\int_{0}^{6} 5 x^{2} + 10 x + 6 d x

\frac{1}{5} (\frac{7 615}{2} - \frac{15}{2} + \frac{1}{2} (ln (75 + 246) - ln (5 + 6)))

Notación decimal

54.09553 \dots

Pasos de solución

\int_{0}^{6} 5 x^{2} + 10 x + 6 d x

Completar el cuadrado

5 x^{2} + 10 x + 6 : 5 (x + 1)^{2} + 1

= \int_{0}^{6} 5 (x + 1)^{2} + 1 d x

Aplicar integración por sustitución

: \int_{1}^{7} 5 u^{2} + 1 d u

= \int_{1}^{7} 5 u^{2} + 1 d u

Aplicar integración por sustitución trigonométrica

: \int_{a r c t a n (5)}^{a r c t a n (5 \cdot 7)} \frac{1}{5} sec^{3} (v) d v

= \int_{a r c t a n (5)}^{a r c t a n (5 \cdot 7)} \frac{1}{5} sec^{3} (v) d v

Sacar la constante:

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

= \frac{1}{5} \cdot \int_{a r c t a n (5)}^{a r c t a n (5 \cdot 7)} sec^{3} (v) d v

Aplicar la reducción de integrales

= \frac{1}{5} [\frac{sec ^{2} ( v ) sin ( v )}{2}]_{a r c t a n (5)}^{a r c t a n (5 \cdot 7)} + \frac{1}{2} \cdot \int_{a r c t a n (5)}^{a r c t a n (5 \cdot 7)} sec (v) d v

\int_{a r c t a n (5)}^{a r c t a n (5 \cdot 7)} sec (v) d v = ln (75 + 246) - ln (5 + 6)

= \frac{1}{5} [\frac{sec ^{2} ( v ) sin ( v )}{2}]_{a r c t a n (5)}^{a r c t a n (5 \cdot 7)} + \frac{1}{2} (ln (75 + 246) - ln (5 + 6))

Simplificar

\frac{1}{5} [\frac{sec ^{2} ( v ) sin ( v )}{2}]_{a r c t a n (5)}^{a r c t a n (5 \cdot 7)} + \frac{1}{2} (ln (75 + 246) - ln (5 + 6)) : \frac{1}{5} ([\frac{1}{2} sec (v) tan (v)]_{a r c t a n (5)}^{a r c t a n (75)} + \frac{1}{2} (ln (75 + 246) - ln (5 + 6)))

= \frac{1}{5} ([\frac{1}{2} sec (v) tan (v)]_{a r c t a n (5)}^{a r c t a n (75)} + \frac{1}{2} (ln (75 + 246) - ln (5 + 6)))

Calcular los limites:

\frac{7 615}{2} - \frac{15}{2}

= \frac{1}{5} (\frac{7 615}{2} - \frac{15}{2} + \frac{1}{2} (ln (75 + 246) - ln (5 + 6)))

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

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