x∧2/根号(1-x∧2)的积分
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x∧2/根号(1-x∧2)的积分
![x∧2/根号(1-x∧2)的积分](/uploads/image/z/11938953-57-3.jpg?t=x%E2%88%A72%2F%E6%A0%B9%E5%8F%B7%281-x%E2%88%A72%29%E7%9A%84%E7%A7%AF%E5%88%86)
设 x = sint,则 dx = cost*dt
∫x^2/√(1-x^2) *dx
=∫(sint)^2 *(cost)*dt/cost
=∫(sint)^2 *dt
=1/2*∫2(sint)^2 *dt
=1/2*∫(1-cos2t)*dt
=1/2*∫dt - 1/2*∫cos2t*dt
=1/2*t - 1/4*sin2t + C
=1/2*arcsinx - 1/2*sint*cost + C
=1/2*arcsinx - 1/2*x*√(1-x^2) + C
∫x^2/√(1-x^2) *dx
=∫(sint)^2 *(cost)*dt/cost
=∫(sint)^2 *dt
=1/2*∫2(sint)^2 *dt
=1/2*∫(1-cos2t)*dt
=1/2*∫dt - 1/2*∫cos2t*dt
=1/2*t - 1/4*sin2t + C
=1/2*arcsinx - 1/2*sint*cost + C
=1/2*arcsinx - 1/2*x*√(1-x^2) + C