What is the integral of $int sin^5x*cos^7x$ ?

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Answer 1 · 2016-08-25T09:10:48

$-(1/8cos^8 x-2/10 cos^10 x + 1/12 cos^12 x) + C$

$sin^5x*cos^7x = sin^4x cos^7x sinx = (1-cos^2 x)^2cos^7x sin x$
$=(1-2cos^2x+cos^4 x)cos^7 x sin x$
$=(cos^7x-2 cos^9 x+cos^11 x)sinx$ so

$int sin^5x*cos^7x dx = int (cos^7x-2 cos^9 x+cos^11 x)sinx dx$
$=-(1/8cos^8 x-2/10 cos^10 x + 1/12 cos^12 x) + C$

or simplifying

$-1/480(27 - 28 cos(2 x) + 5 cos(4 x)) cos^8x$

What is the integral of int sin^5x*cos^7x int sin^5x*cos^7x ?