求不定积分 sin^4x dx?

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求不定积分 sin^4x dx?
求不定积分 sin^4x dx?

求不定积分 sin^4x dx?
∫sin^4x dx
=∫(1-cos^2x )sin^2xdx
=∫sin^2xdx-1/4∫(sin2x)^2dx
=1/2∫(1-cos2x)dx-1/8∫(1-cos4x)dx
=1/2x-1/2sin2x-1/8x+1/4sin4x+C
=3/8x-1/2sin2x+1/4sin4x+C

(sinx) ^4 = (1-cos 2x)^2 /4 = ( 1-2cos2x+ (cos2x)^2 ) /4
= 3/8 – cos2x /2 + cos4x /8
∫(sinx) ^4 dx = ∫( 3/8 – cos2x /2 + cos4x /8 ) dx
= 3x /8 – sin2x /4 + sin4x /32 +C

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