原标题：不定积分∫cos^4(4x-1/2)dx的计算

主要内容：

本内容用凑分法、分部积分法以及三角公式变形等有关知识，介绍不定积分∫cos^4(4x-1/2)dx的计算步骤。

详细步骤：

I=∫cos^4(4x-1/2)dx

对微元进行凑分

=(1/4)∫cos^4(4x-1/2)d(4x-1/2)

=(1/4)∫cos^3(4x-1/2)cos(4x-1/2)d(4x-1/2)

对三角函数进行凑分

=(1/4)∫cos^3(4x-1/2)dsin(4x-1/2)

以下进行分部积分法

=(1/4)cos^3(4x-1/2)sin(4x-1/2)-(1/4)∫sin(4x-1/2)dcos^3(4x-1/2)；

=(1/4)cos^3(4x-1/2)sin(4x-1/2)+3∫sin^2(4x-1/2)cos^2(4x-1/2)dx；

使用sin^2x+cos^2x=1公式进行变形，

=(1/4)cos^3(4x-1/2)sin(4x-1/2)+3∫[1-cos^2(4x-1/2)]cos^2(4x-1/2)dx

=(1/4)cos^3(4x-1/2)sin(4x-1/2)+3∫cos^2(4x-1/2)dx-3I,则：

4I=(1/4)cos^3(4x-1/2)sin(4x-1/2)+3∫cos^2(4x-1/2)dx;