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ahrayia [7]
3 years ago
9

Haya la integral de xarcsenx/((1-x^2)^1/2). Utiliza la sustitución de t= arcsenx

Mathematics
1 answer:
ikadub [295]3 years ago
7 0

Answer:

<em>Explicación más abajo</em>

Step-by-step explanation:

Integración Indefinida

La integral

I=\int \dfrac{x .arcsen\left(x\right)}{\sqrt{1-x^2}}

Se resuelve con el cambio de variables:

t=arcsen(x)

Una vez hechos los cambios, la integral se resuelve en función de t:

I=sen(t)-t.cos(t)+C

Hay que devolver los cambios para mostrarla en función de x.

El cambio de variables también se puede escribir:

x=sen(t)

Recordando que

cos(t)=\sqrt{1-sen^2(t)}

Entonces:

cos(t)=\sqrt{1-x^2}

Devolviendo los cambios:

I=sen(t)-t.cos(t)+C=x-arcsen(x)\sqrt{1-x^2}+C

Es la respuesta correcta

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