www.5615.net > ∫xArCtAnxDx0到1的定积分

∫xArCtAnxDx0到1的定积分

∫xarctanxdx=∫arctanxd(x/2)=(x/2)arctanx-∫x/2darctanx=(x/2)arctanx-(1/2)∫x/(1+x)dx =(x/2)arctanx-(1/2)∫[1-1/(1+x)]dx=(x/2)arctanx-(1/2)(x-arctanx)=(1/2)(xarctanx+arctanx-x)|(0~1) =(1/2)(π/4+π/4-1)=π/4-1/2

∫xarctanxdx=∫arctanxd(x??/2)=(x??/2)arctanx-∫x??/2darctanx=(x??/2)arctanx-(1/2)∫x??/(1+x??)dx=(x??/2)arctanx-(1/2)∫[1-1/(1+x??)]dx=(x??/2)arctanx-(1/2)(x-arctanx)=(1/2)(x??arctanx+arctanx-x)|(0~1)=(1/2)(π/4+π/4-1)=π/4-1/2

很简单,分步积分法 ∫xarctanxdx=x^2*arctanx/2-∫1/2*x^2/(1+x^2)dx=x^2*arctanx/2-x/2-arctanx/2 带0,1入得π/4-1/2

∫xarctanxdx=1/2∫arctanxdx=1/2xarctanx-1/2∫x/(1+x)dx=1/2xarctanx-1/2∫[1-1/(1+x)]dx=1/2xarctanx-1/2x+1/2arctanx+c

∫xarctanxdx=∫arctanxd(x^2/2)=(1/2)(x^2)arctanx-(1/2)∫[x^2/(1+x^2)]dx=(1/2)(x^2)arctanx-(1/2)∫[x^2+1-1]/(1+x^2)dx=(1/2)(x^2)arctanx-(1/2)∫[1-1/(1+x^2)]dx=(1/2)(x^2)arctanx-(1/2)x+(1/2)arctanx x从0到1=(1/2)arctan1-1/2+(1/2)arctan1-(1/2)arctan0=π/4-1/2

∫xarctanxdx=x/2arctanx-1/2x+1/2arctanx+c.c为积分常数.解答过程如下:∫xarctanxdx=∫arctanxdx/2=x/2arctanx-∫x/2darctanx=x/2arctanx-1/2∫x/(1+x)dx=x/2arctanx-1/2∫(x+1-1)/(1+x)dx=x/2arctanx-1/2∫1-1/(1+x)dx=x/2

∫ x * arctanx dx= ∫ arctanx d(x/2)= (x/2)arctanx - (1/2)∫ x d(arctanx)= (x/2)arctanx - (1/2)∫ x/(x + 1) dx= (x/2)arctanx - (1/2)∫ (x + 1 - 1)/(x + 1) dx= (x/2)arctanx - (1/2)∫ dx + (1/2)∫ dx/(x + 1)= (x/2)arctanx - x/2 + (1/2)arctanx + c

∫xarctanxdx=1/2∫arctanx*2xdx=1/2∫arctanxdx^2=1/2xarctanx-1/2∫x^2*1/(x^2+1)dx=1/2xarctanx-1/2∫(x^2+1-1)dx/(x^2+1)=1/2xarctanx-1/2∫dx+1/2∫dx/(x^2+1)=1/2xarctanx-x/2+1/2*arctanx+C=1/2*(xarctanx-x+arctanx)+C

∫xarctanxdx 分部积分=(∫arctanxdx^2)/2=x^2arctanx|(0,1)/2 - ∫x^2darctanx/2=π/8 - ∫(x^2/1+x^2)dx/2=π/8 - ∫(1-1/(1+x^2))dx/2=π/8 - ∫dx/2 + ∫dx/(1+x^2)/2=π/8 - x/2|(0,1) + arctanx/2|(0,1)=π/8 - 1/2 + π/8=π/4 - 1/2 X_Q_T提醒的没错,而且我把分部积分的公式记错了,汗啊

网站地图

All rights reserved Powered by www.5615.net

copyright ©right 2010-2021。
www.5615.net内容来自网络,如有侵犯请联系客服。zhit325@qq.com