To solve the integral \(\int(\operatorname{arcsen} x)^{2} d x\), we can use integration by parts. Integration by parts is based on the formula \(\int u \, dv = uv - \int v \, du\). We will choose \(u = (\operatorname{arcsen} x)^{2}\) and \(dv = dx\). Then, we will differentiate \(u\) to find \(du\) and integrate \(dv\) to find \(v\). Finally, we will apply the integration by parts formula to solve the integral.
Queremos resolver la integral \(\int (\operatorname{arcsen} x)^{2} \, dx\). Utilizaremos la técnica de integración por partes, donde elegimos \(u = (\operatorname{arcsen} x)^{2}\) y \(dv = dx\).
Calculamos \(du\) y \(v\):
- \(du = 2 \cdot \operatorname{arcsen}(x) \cdot \frac{1}{\sqrt{1 - x^{2}}} \, dx\)
- \(v = x\)
Aplicamos la fórmula de integración por partes:
\[
\int u \, dv = uv - \int v \, du
\]
Sustituyendo los valores:
\[
\int (\operatorname{arcsen} x)^{2} \, dx = x \cdot (\operatorname{arcsen} x)^{2} - \int x \cdot \left(2 \cdot \operatorname{arcsen}(x) \cdot \frac{1}{\sqrt{1 - x^{2}}}\right) \, dx
\]
El resultado de la integral se simplifica a:
\[
x \cdot (\operatorname{arcsen} x)^{2} - 2x + 2\sqrt{1 - x^{2}} \cdot \operatorname{arcsen}(x) + C
\]
donde \(C\) es la constante de integración.
\[
\boxed{\int (\operatorname{arcsen} x)^{2} \, dx = x \cdot (\operatorname{arcsen} x)^{2} - 2x + 2\sqrt{1 - x^{2}} \cdot \operatorname{arcsen}(x) + C}
\]
Answered
