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What is integral `int sin^3x*cos^3x dx`

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What is integral `int sin^3x*cos^3x dx`

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The integral `int sin^3x*cos^3x dx` has to be determined.

`int sin^3x*cos^3x dx`

=> `int sin^3x*(1 - sin^2x)*cos x dx`

=> `int sin^3x*cos x dx - int sin^5x*cos x dx`

let `sin x = y => dy = cos x*dx`

=> `int y^3 dy - int y^5 dy`

=> `y^4/4 - y^6/6`

substitute `y = sin x`

=> `(sin^4x)/4 - (sin^6x)/6 + C`

The integral `int sin^3x*cos^3x dx = (sin^4x)/4 - (sin^6x)/6 + C`

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