integral de x^2sqrt(a^2+a^2x^2)

Solución

\int x^{2} a^{2} + a^{2} x^{2} d x

a (- \frac{1}{4} (\frac{1}{2} x 1 + x^{2} + \frac{1}{2} ln x + 1 + x^{2}) + \frac{1}{4} x (x^{2} + 1)^{\frac{3}{2}}) + C

Pasos de solución

\int x^{2} a^{2} + a^{2} x^{2} d x

a^{2} + a^{2} x^{2} = a^{2} 1 + x^{2}

= \int x^{2} a^{2} 1 + x^{2} d x

Sacar la constante:

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

= a^{2} \cdot \int x^{2} 1 + x^{2} d x

Aplicar integración por sustitución trigonométrica

: \int sec^{3} (u) tan^{2} (u) d u

= a^{2} \cdot \int sec^{3} (u) tan^{2} (u) d u

Re-escribir usando identidades trigonométricas

= a^{2} \cdot \int sec^{3} (u) (- 1 + sec^{2} (u)) d u

Expandir

sec^{3} (u) (- 1 + sec^{2} (u)) : - sec^{3} (u) + sec^{5} (u)

= a^{2} \cdot \int - sec^{3} (u) + sec^{5} (u) d u

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

= a^{2} (- \int sec^{3} (u) d u + \int sec^{5} (u) d u)

\int sec^{3} (u) d u = \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

\int sec^{5} (u) d u = \frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

= a^{2} (- (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣) + \frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣))

Sustituir en la ecuación

u = arctan (x)

= a^{2} (- (\frac{1}{2} sec (arctan (x)) tan (arctan (x)) + \frac{1}{2} ln ∣ tan (arctan (x)) + sec (arctan (x)) ∣) + \frac{sec ^{4} ( arctan ( x ) ) sin ( arctan ( x ) )}{4} + \frac{3}{4} (\frac{1}{2} sec (arctan (x)) tan (arctan (x)) + \frac{1}{2} ln ∣ tan (arctan (x)) + sec (arctan (x)) ∣))

Simplificar

a^{2} (- (\frac{1}{2} sec (arctan (x)) tan (arctan (x)) + \frac{1}{2} ln ∣ tan (arctan (x)) + sec (arctan (x)) ∣) + \frac{sec ^{4} ( arctan ( x ) ) sin ( arctan ( x ) )}{4} + \frac{3}{4} (\frac{1}{2} sec (arctan (x)) tan (arctan (x)) + \frac{1}{2} ln ∣ tan (arctan (x)) + sec (arctan (x)) ∣)) : a (- \frac{1}{4} (\frac{1}{2} x 1 + x^{2} + \frac{1}{2} ln x + 1 + x^{2}) + \frac{1}{4} x (x^{2} + 1)^{\frac{3}{2}})

= a (- \frac{1}{4} (\frac{1}{2} x 1 + x^{2} + \frac{1}{2} ln x + 1 + x^{2}) + \frac{1}{4} x (x^{2} + 1)^{\frac{3}{2}})

Agregar una constante a la solución

= a (- \frac{1}{4} (\frac{1}{2} x 1 + x^{2} + \frac{1}{2} ln x + 1 + x^{2}) + \frac{1}{4} x (x^{2} + 1)^{\frac{3}{2}}) + C

\frac{d}{d x} (sec (x) + tan (x))

\frac{13}{x}

f (x) = 2 x^{4} - 6 x^{2}, at x = 1

x y 2 = 2 + x y

\int_{0}^{1} 2 x^{2} d x