sec ^5x微积分

(secx)^5
【求导】
[(secx)^5]'
= 5(secx)^4 *(secxtgx)
= 5(secx)^5 tgx
【积分】
∫(secx)^5 dx
=∫(secx)^3 d(tgx)
=(secx)^3 (tgx) -∫3 (secx)^2 (secx tgx) (tgx) dx
=(secx)^3 (tgx) -∫3 (secx)^3 (tgx)^2 dx
=(secx)^3 (tgx) -∫3 (secx)^3 [(secx)^2-1] dx
=(secx)^3 (tgx) - 3∫ (secx)^5 - (secx)^3 dx
所以4∫(secx)^5 dx = (secx)^3 (tgx) +3∫(secx)^3 dx
同理 ∫(secx)^3 dx
= ∫(secx) d(tgx)
= secxtgx -∫(secx tgx) (tgx) dx
= secxtgx -∫(secx) (tgx)^2 dx
= secxtgx -∫(secx) [(secx)^2-1] dx
= secxtgx -∫(secx)^3 - (secx) dx
2∫(secx)^3 dx = secxtgx +∫secx dx
所以
∫(secx)^5 dx
= (secx)^3 (tgx) /4 +（3/4）∫(secx)^3 dx
= (secx)^3 (tgx) /4 +（3/4）（1/2）(secxtgx +∫secx dx)
= (secx)^3 (tgx) /4 +（3/8）secxtgx + (3/8)∫secx dx
= (secx)^3 (tgx) /4 +（3/8）secxtgx + (3/8) ln ｜secx + tgx｜+ C