Verified answer

Hola,

∫ x arcsenx dx =

sea:

x dx = dv → [1/(1+1)] x^(1+1) = (1/2)x² = v

arcsenx = u → [1 /√(1 - x²)] dx = du

luego, integrando por partes:

∫ u dv = v u - ∫ v du

∫ arcsenx x dx = (1/2)x² arcsenx - ∫ (1/2)x² [1 /√(1 - x²)] dx =

reescribamosla como:

(1/2)x² arcsenx + (1/2) ∫ [- x² /√(1 - x²)] dx (#)

cuanto a la restante integral, sumemos y sustraigamos 1 al numerador:

∫ [- x² /√(1 - x²)] dx =

∫ [(1 - x²) - 1] /√(1 - x²)} dx =

luego, partiendo y simplificando:

∫ [(1 - x²) /√(1 - x²)] dx - ∫ [1 /√(1 - x²)] dx =

∫ √(1 - x²) dx - ∫ [1 /√(1 - x²)] dx =

(siendo la segunda integral inmediata)

∫ √(1 - x²) dx - arcsenx =

ahora integremos la restante integral por partes, poniendo:

dx = dv → x = v

√(1 - x²) = u → (1 - x²)^(1/2) = u → (1/2)(- 2x)(1 - x²)^[(1/2) -1] dx =

- x (1 - x²)^(-1/2) dx = [- x /√(1 - x²)] dx = du

y obteniendo:

∫ u dv = v u - ∫ v du

∫ √(1 - x²) dx - arcsenx = {x √(1 - x²) - ∫ x [- x /√(1 - x²)] dx} - arcsenx =

x √(1 - x²) - ∫ [- x² /√(1 - x²)] dx - arcsenx

luego, resumiendo (ve sobre):

∫ [- x² /√(1 - x²)] dx = x √(1 - x²) - ∫ [- x² /√(1 - x²)] dx - arcsenx

teniendo la misma integral en ambos miembros, recojamosla al primer miembro:

∫ [- x² /√(1 - x²)] dx + ∫ [- x² /√(1 - x²)] dx = x √(1 - x²) - arcsenx + C

2 ∫ [- x² /√(1 - x²)] dx = x √(1 - x²) - arcsenx + C

por tanto:

∫ [- x² /√(1 - x²)] dx = (1/2) [x √(1 - x²) - arcsenx] + C = (1/2)x √(1 - x²) -

(1/2)arcsenx + C

entonces la expresión anterior (#) se vuelve:

(1/2)x² arcsenx + (1/2) ∫ [- x² /√(1 - x²)] dx = (1/2)x² arcsenx +

(1/2) [(1/2)x √(1 - x²) - (1/2)arcsenx] + C =

(1/2)x² arcsenx + (1/4)x √(1 - x²) - (1/4)arcsenx + C

en conclusión:

∫ x arcsenx dx = (1/2)x² arcsenx + (1/4)x √(1 - x²) - (1/4)arcsenx + C

espero haber sido de ayuda

¡Saludos!

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∫x*arcsin(x) dx

Use integration by parts:

u = arcsin(x)

du = dx/√1 - x²

dv = x dx

v = x²/2

uv - ∫v du

x²*arcsin(x)/2 - 1/2*∫x²/√1-x² dx

Substituting x = sin(t) in the integral.

x²*arcsin(x)/2 - 1/2*∫sin²(t) dt

x²*arcsin(x)/2 - 1/4[∫1 - cos(2t) dt]

x²*arcsin(x)/2 - t/4 + sin(t)*cos(t)/4 + C

x²*arcsin(x)/2 - arcsin(x)/4 + (x√1 - x²)/4 + C