Question

求cosθ^4的积分

Answer 1

求不定积分∫cos⁴θdθ
解：cos⁴θ=(cos²θ)²=[(1+cos2θ)/2]²=(1/4)(1+2cos2θ+cos²2θ)=(1/4)[1+2cos2θ+(1+cos4θ)/2]
=(1/8)(3+4cos2θ+cos4θ)
故原式=(1/8)∫(3+4cos2θ+cos4θ)dθ=(1/8)[∫3dθ+2∫cos2θd(2θ)+(1/4)∫cos4θd(4θ)]
=(1/8)[3θ+2sin2θ+(1/4)sin4θ]+C

Answer 2

积分(cosθ)^4dθ
=积分cos^3 θ dsinθ
=sinθ cos^3 θ + 积分sinθ * 3 cos^2 θ sinθ dθ
=sinθ cos^3 θ + 3/4 sin^2 (2θ) dθ
=sinθ cos^3 θ + 3/8 (1-cos(4θ))dθ
=sinθ cos^3 θ + 3/8 θ - 3/32 sin4θ + C

Answer 3

(cosθ)^4还是cos(θ^4)?

追问

(cosθ)^4

追答

1/4{(sinθ)*(cosθ)^3+(3/2)[(sinθ)(cosθ)+θ]}+c