Question

求解正割函数的五次方的不定积分的全过程？

正割函数，五次方，不定积分

Answer 1

K = ∫secx dx = ∫secx(secx+tanx)/(secx+tanx) dx = ∫(secxtanx+sec�0�5x)/(secx+tanx) dx
= ∫1/(secx+tanx) d(secx+tanx) = ln|secx+tanx|
I = ∫sec�0�6x dx
= ∫secx d(tanx)
= secx*tanx - ∫tanx d(secx)
= secx*tanx - ∫tanx*(secx*tanx) dx
= secx*tanx - ∫tan�0�5x*secx dx
= secx*tanx - ∫(sec�0�5x-1)*secx dx
= secx*tanx - I + ∫secx dx
2I = secx*tanx + ∫secx dx
I = (1/2)secx*tanx + (1/2)ln|secx+tanx|

J = ∫sec^5x dx
= ∫sec�0�6x d(tanx)
= sec�0�6x*tanx - ∫tanx d(sec�0�6x)，integration by parts
= sec�0�6x*tanx - ∫tanx*(3sec�0�5x*secxtanx) dx
= sec�0�6x*tanx - 3∫tan�0�5x*sec�0�6x dx
= sec�0�6x*tanx - 3∫(sec�0�5x-1)*sec�0�6x dx
= sec�0�6x*tanx - 3∫(sec^5x - sec�0�6x) dx
= sec�0�6x*tanx - 3J + 3∫sec�0�6x dx
4J = sec�0�6x*tanx + 3I
J = (1/4)sec�0�6x*tanx + (3/4)[(1/2)secx*tanx + (1/2)ln|secx+tanx|]

所以答案 = (1/4)sec�0�6(x)tan(x) + (3/8)sec(x)tan(x) + (3/8)ln|sec(x)+tan(x)| + C

Answer 2

K1=∫secxdx
=∫secx(secx+tanx)/(secx+tanx)dx
=∫d(secx+tanx)/(ecx+tanx)
=ln|secx+tanx|+C
k2=∫sec^5xdx=∫sec^3xdtanx=tanxsec^3x-∫tanxdsec^3x=tanxsec^3x-3∫tan^2xsec^3dx=tanxsec^3x-3k2+3∫sec^3xdx即：k2=(1/4)tanxsec^3x+(3/4)∫sec^3xdx k3=∫sec^3xdx=∫secxdtanx=secxtanx-∫tanxdsecx=secxtanx-∫(1-cos^2x)dx/cos^3x=secxtanx-k3+∫secxdx=secxtanx-k3+k1k3=(1/2)sectanx+(1/2)ln|secx+tanx|+c 所以k2=(1/4)tanxsec^3x+(3/8)secxtanx+(3/8)ln|secx+tanx|+c.