搜狗做题网求sinx的三次方除以(sinx+cosx)的不定积分
来源：学生作业帮 编辑：搜狗做题网作业帮 分类：数学作业 时间：2024/04/28 05:18:20
求sinx的三次方除以(sinx+cosx)的不定积分
∫ (sinx)^3/(cosx+sinx) dx
=1/√2*∫ (sinx)^3/(sin45*cosx+cos45*sinx) dx
=1/√2*∫ (sinx)^3/sin(45+x) dx
设45+x=t ∴dx/dt=1有
=1/√2*∫ (sin(t-45))^3/sint dx
=1/√2*∫ (sint /√2 - cost /√2 )^3/sint dx 将（a-b）^3形式多项式分解
=1/√2*∫ (sint^3 /2√2 - 3*sint^2 /2 *cost /√2 + 3*cost^2 /2 *sint /√2 - cost^3 /2√2 )/sint dx
=1/√2*∫ (sint^3 /2√2 - 3/2√2*sint^2 *cost + 3√2*cost^2 *sint - cost^3 /2√2 )/sint dx
=1/4*∫ (sint^3 - 3*sint^2 *cost + 3*cost^2 *sint - cost^3 )/sint dx
=1/4*∫sint^2 - 3*sint*cost + 3*cost^2 - cost^3/sint dx
=1/4*∫{1 - 3/2*sin2t + 2*cost^2 } dt - 1/4∫cost^2/sint d sint
=1/4*∫{2 - 3/2*sin2t + cos2t } dt - 1/4∫(1-sint^2)/sint d sint
=t/2 + 1/4*3/4*cos2t + 1/8*sin2t - 1/4∫1/sint-sint d sint
=t/2 + 3/16*cos2t + 1/8*sin2t - 1/4*ln|sint| + sint^2/8 + C
=1/√2*∫ (sinx)^3/(sin45*cosx+cos45*sinx) dx
=1/√2*∫ (sinx)^3/sin(45+x) dx
设45+x=t ∴dx/dt=1有
=1/√2*∫ (sin(t-45))^3/sint dx
=1/√2*∫ (sint /√2 - cost /√2 )^3/sint dx 将（a-b）^3形式多项式分解
=1/√2*∫ (sint^3 /2√2 - 3*sint^2 /2 *cost /√2 + 3*cost^2 /2 *sint /√2 - cost^3 /2√2 )/sint dx
=1/√2*∫ (sint^3 /2√2 - 3/2√2*sint^2 *cost + 3√2*cost^2 *sint - cost^3 /2√2 )/sint dx
=1/4*∫ (sint^3 - 3*sint^2 *cost + 3*cost^2 *sint - cost^3 )/sint dx
=1/4*∫sint^2 - 3*sint*cost + 3*cost^2 - cost^3/sint dx
=1/4*∫{1 - 3/2*sin2t + 2*cost^2 } dt - 1/4∫cost^2/sint d sint
=1/4*∫{2 - 3/2*sin2t + cos2t } dt - 1/4∫(1-sint^2)/sint d sint
=t/2 + 1/4*3/4*cos2t + 1/8*sin2t - 1/4∫1/sint-sint d sint
=t/2 + 3/16*cos2t + 1/8*sin2t - 1/4*ln|sint| + sint^2/8 + C