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Popular >

integral de 0 a 2 de sqrt(1+(2-2x)^2)

Solución

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

Solución

5 + \frac{1}{4} ln (2 + 5) - \frac{1}{4} ln (- 2 + 5)

+1

Notación decimal

2.95788 \dots

Pasos de solución

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

Expandir

1 + (2 - 2 x)^{2} : 4 x^{2} - 8 x + 5

= \int_{0}^{2} 4 x^{2} - 8 x + 5 d x

Completar el cuadrado

4 x^{2} - 8 x + 5 : 4 (x - 1)^{2} + 1

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

Aplicar integración por sustitución

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

= \int_{- 1}^{1} 4 u^{2} + 1 d u

Aplicar integración por sustitución trigonométrica

: \int_{- a r c t a n (2)}^{a r c t a n (2)} \frac{1}{2} sec^{3} (v) d v

= \int_{- a r c t a n (2)}^{a r c t a n (2)} \frac{1}{2} sec^{3} (v) d v

Sacar la constante:

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

= \frac{1}{2} \cdot \int_{- a r c t a n (2)}^{a r c t a n (2)} sec^{3} (v) d v

Aplicar la reducción de integrales

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

\int_{- a r c t a n (2)}^{a r c t a n (2)} sec (v) d v = ln (2 + 5) - ln (- 2 + 5)

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

Simplificar

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

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

Calcular los limites:

25

= \frac{1}{2} (25 + \frac{1}{2} (ln (2 + 5) - ln (- 2 + 5)))

Simplificar

= 5 + \frac{1}{4} ln (2 + 5) - \frac{1}{4} ln (- 2 + 5)

Ejemplos populares

integral de 0 a pi de (3+cos(x))

\int_{0}^{π} (3 + cos (x)) d x

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

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

integral de pi/4 a pi/3 de sin^2(2x)

\int_{\frac{π}{4}}^{\frac{π}{3}} sin^{2} (2 x) d x

integral de 0 a x de sec^2(x)

\int_{0}^{x} sec^{2} (x) d x

integral de (y^2)/(40) a 20 de (40+x)

\int_{\frac{y ^{2}}{40}}^{20} (40 + x) d x