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JEE MainMathematicsModerateIndefinite integration

If C arbitrary constant then the value of
∫ 4sinxcosx2cos3x2\int\ 4sinxcos\frac{x}{2}cos\frac{3x}{2} dx is equal to.

Show answer and solution
∫ 4sinxcosx2cos3x2dx=∫ 2sinx(cos2x+cosx)dx\int\ 4sinxcos\frac{x}{2}cos\frac{3x}{2}dx = \int\ 2sinx(cos2x + cosx)dx
=∫ 2sinxcos2xdx+∫ sin2xdx= \int\ 2sinxcos2xdx + \int\ sin2xdx
=∫ (sin3x−sinx+sin2x)dx=cosx−12cos2x−13cos3x+c= \int\ (sin3x - sinx + sin2x)dx = cosx - \frac{1}{2}cos2x - \frac{1}{3}cos3x + c

Source: Grass-G / M-12 Mathematics Paper 2 Nov.docx

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