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intergrating trig watch

1. how do i intergrate sin^2 2x*cos 2x ?

thanks alot
2. (Original post by sarah12345)
how do i intergrate sin^2 x*cos 2x ?

thanks alot
Write (sinx)^2 in terms of (cos2x) then also write any newly generated terms in (cos2x)^2 as expressions in terms of (cos4x) so that you have an integral without squares of trigonometric ratios. You then know the standard result of integrating CosAx.
3. int sin^2 x*cos 2x dx = int (1-cos²x) cos2x dx

cos²x = (cos2x +1 )/2

int (1-[(cos2x +1 )/2]) cos2x dx =
int (½ - ½cos2x ) cos2x dx =
int ½ cos2x - ½cos²2x dx =
int ½ cos2x - ¼ (cos4x +1 ) dx =
int ½ cos2x - ¼cos4x - ¼ dx =
¼ sin2x - 1/16 sin4x - x/4 + C

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