integration of sin8x dx /(9+sin⁴x)^1/2

plz explain in detail

take sin²x=t

dt=2sinxcosxdx=sin2xdx

now the numerator of the expression: sin8xdx=2sin4xcos4xdx=4sin2xcos2xcos4xdx=4cos2xcos4xdt;

now we know that cos2x=cos²x-sin²x=1-2sin²x=1-2t

similarly, cos4x=1-sin²2x=1-4sin²xcos²x=1-4t(1-t)=1-4t+4t²=4t²-2t-2t+1=(2t-1)²

the denominator is (9+t²)^1/2

put the expressions in position

and expand the numerator

you will get different powers of t

separate than integrate each part separately

Cool

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