形如∫x^3cos^3(x+a)dx不定积分计算详细过程A1

eq\i(,,x3cos3(x+3))dx不定积分计算主要内容:本文通过不定积分凑分、分部积分法等知识内容，介绍不定积分eq\i(,,x3cos3(x+3))dx的计算过程和主要步骤。主要步骤：I=eq\i(,,x3cos3(x+3))dx将积分函数余弦部分拆解并凑分，步骤如下：I=eq\i(,,x3cos2(x+3)*cos(x+3)d(x+3))=eq\i(,,x3cos2(x+3)dsin(x+3)),利用公式sin2(x+3)+cos2(x+3)=1,变形得：I=eq\i(,,x3[1-sin2(x+3)]dsin(x+3))将积分部分进行拆解，得：I=eq\i(,,x3dsin(x+3))-eq\i(,,x3sin2(x+3)dsin(x+3))=eq\i(,,x3dsin(x+3))-eq\f(1,3)eq\i(,,x3dsin3(x+3))对两个不定积分进行分部积分，有：I=x3sin(x+3)-eq\i(,,sin(x+3)dx3)-eq\f(1,3)x3sin3(x+3)+eq\f(1,3)eq\i(,,sin3(x+3)dx3)，继续进行变形及凑分，有：=x3sin(x+3)-3eq\i(,,x2sin(x+3)dx)-eq\f(1,3)x3sin3(x+3)+eq\i(,,x2sin3(x+3)dx).=x3sin(x+3)-A-eq\f(1,3)x3sin3(x+3)+B.其中：A=3eq\i(,,x2sin(x+3)dx)=3eq\i(,,x2sin(x+3)d(x+3))=-3eq\i(,,x2dcos(x+3)),本步骤仍为凑分法。=-3x2cos(x+3)+3eq\i(,,cos(x+3)dx2)=-3x2cos(x+3)+6eq\i(,,xcos(x+3)dx)=-3x2cos(x+3)+6eq\i(,,xdsin(x+3))=-3x2cos(x+3)+6xsin(x+3)-6eq\i(,,sin(x+3)dx)=-3x2cos(x+3)+6xsin(x+3)+6cos(x+3)。B=eq\i(,,x2sin3(x+3)dx),将积分函数正弦部分拆解并凑分，步骤如下：=eq\i(,,x2sin2(x+3)sin(x+3)d(x+3))=-eq\i(,,x2sin2(x+3)dcos(x+3))=-eq\i(,,x2[1-cos2(x+3)]dcos(x+3))=-eq\i(,,x2dcos(x+3))+eq\i(,,x2cos2(x+3)dcos(x+3))=-x2cos(x+3)+2eq\i(,,xcos(x+3)dx)+eq\f(1,3)eq\i(,,x2dcos3(x+3))=-x2cos(x+3)+2eq\i(,,xdsin(x+3))+eq\f(1,3)x2cos3(x+3)-eq\f(1,3)eq\i(,,xcos2(x+3)dsin(x+3))=-x2cos(x+3)+2xsin(x+3)-2eq\i(,,sin(x+3)dx)+eq\f(1,3)*x2cos3(x+3)-eq\f(2,3)eq\i(,,xdsin(x+3))+eq\f(2,3)eq\i(,,xsin2(x+3)dsin(x+3))=-x2cos(x+3)+2xsin(x+3)-2eq\i(,,sin(x+3)d(x+3))+eq\f(1,3)*x2cos3(x+3)-eq\f(2,3)xsin(x+3)+eq\f(2,3)eq\i(,,sin(x+3)dx)+eq\f(2,9)eq\i(,,xdsin3(x+3))=-x2cos(x+3)+2xsin(x+3)+2cos(x+3)+eq\f(1,3)x2cos3(x+3)-eq\f(2,3)xsin(x+3)-eq\f(2,3)cos(x+3)+eq\f(2,9)∫xdsin3(x+3)=-x2cos(x+3)+eq\f(4,3)xsin(x+3)+eq\f(4,3)cos(x+3)+eq\f(1,3)x2cos3(x+3)+eq\f(2,9)xsin3(x+3)-eq\f(2,9)∫sin3(x+3)d(x+3)=-x2cos(x+3)+eq\f(4,3)xsin(x+3)+eq\f(4,3)cos(x+3)+eq\f(1,3)x2cos3(x+3)+eq\f(2,9)xsin3(x+3)+eq\f(2,9)cos4(x+3)。综上所求不定积分为：I=x3sin(x+3)+3x2cos(x+3)-6xsin(x+3)-6cos(x+3)+eq\f(1,3)x3sin3(x+3)-x2cos(x+3)+eq\f(4,3)xsin(x+3)+eq\f(4,3)cos(x+3)+eq\f(1,3)x2cos3(x+3)+eq\f(2,9)xsin3(x+3)+eq\f(2,9)cos4(x+3)+C，=x3sin(x+3)+2x2cos(x+3)-eq\f(14,3)xsin