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Question:

Find(∫π/40dxcos3x√2sin2x).Find(∫π/40dxcos3x√2sin2x).(∫π/40dxcos3x√2sin2x)(∫π/40dxcos3x√2sin2x)(∫π/40dxcos3x√2sin2x)(∫π/40dxcos3x√2sin2x)((∫π/40dxcos3x√2sin2x∫π/40dxcos3x√2sin2x∫π/40dxcos3x√2sin2x∫π/40dxcos3x√2sin2x∫π/40dxcos3x√2sin2x∫π/40∫∫∫π/40π/4π/4π/4ππ////44000dxcos3x√2sin2xdxcos3x√2sin2xdxcos3x√2sin2xdxcos3x√2sin2xdxdxddxxcos3x√2sin2xcos3x√2sin2xcos3coscoscos333xx√2sin2x√2sin2x√√2sin2x2sin2x22sinsin22xx))?

Solution:

∫π/40dxcos3x√2sin2x=∫π/40dxcos3x√4sinxcosx=∫π/40dxcos4x√4tanx=12∫π/40sec4x√tanxdx=∫π/40sec2x(1+tan2x)√tanxdxt=tanx⇒dt=sec2xdx⇒12∫π/40sec2x(1+tan2x)√tanxdx=12∫10(t󔼔+t32)dt=[t12+t525]10=65