大学高等数学，请教这道不定积分怎么算? 高等数学，请问这道积分怎么算?
\u9ad8\u7b49\u6570\u5b66\uff0c\u8bf7\u6559\u8fd9\u9053\u4e0d\u5b9a\u79ef\u5206\u600e\u4e48\u7b97?\u539f\u5f0f=\u222bx/(x²+1)dx+\u222b1/(x²+1)dx
=1/2\u00b7\u222b1/(x²+1)\u00b7d(x²+1)+arctanx
=1/2\u00b7ln(x²+1)+arctanx+C
\u3010\u9644\u6ce8\u3011
\u222b1/(x²+1)dx=arctanx+C
\u222bx/(x²+1)dx=1/2\u00b7ln(x²+1)+C
\u5982\u56fe\u3002
=1/2ln(x^2-x+1)-∫1/(x-1/2)^2+3/4)dx
=1/2ln(x^2-x+1)-4/3*∫1/(2x/√3-1/√3)^2+1)dx
=1/2ln(x^2-x+1)-2/√3*arctan(2x/√3-1/√3)+C
绛旓細=1/2ln(x^2-x+1)-2/鈭3*arctan(2x/鈭3-1/鈭3)+C
绛旓細涓銆佲垰(1 + x²) = 鈭(1 + tan²y) = 鈭歴ec²y = secy 璁緓 = tany锛宒x = sec²y dy 鈭 x⁵/鈭(1 + x²) dx = 鈭 tan⁵ysec²y/secy dy = 鈭 tan⁵ysecy dy = 鈭 (sec²y - 1)² dsecy = 鈭 (se...
绛旓細褰搙澶т簬0鏃讹紝f'(3x^2)=x^4+x=(3x^2)^2/9+鏍瑰彿锛3x^2锛/鏍瑰彿3,鎵浠'(X)=X^2/9+鏍瑰彿(X/3), f(X)灏辨槸閭ｄ釜寮忓瓙鐨涓嶅畾绉垎锛瑙ｅ緱 f(X)=X^3/27+2鏍瑰彿(X^3)/(3鏍瑰彿3)+C.褰搙<0鏃讹紝f'(3x^2)=x^4+x=(3x^2)^2/9-鏍瑰彿锛3x^2锛/鏍瑰彿3,鎵浠'(X)=X^2/9-鏍...
绛旓細鏂规硶濡備笅锛岃浣滃弬鑰冿細
绛旓細鍘熷紡=鈭玿/(x²+1)dx+鈭1/(x²+1)dx =1/2路鈭1/(x²+1)路d(x²+1)+arctanx =1/2路ln(x²+1)+arctanx+C 銆愰檮娉ㄣ戔埆1/(x²+1)dx=arctanx+C 鈭玿/(x²+1)dx=1/2路ln(x²+1)+C ...
绛旓細鈭(0->蟺) 鈭(1-sinx) dx =鈭(0->蟺) 鈭(sin^2 x/2 +cos^2 x/2 -2sinx/2 * cosx/2) dx =鈭(0->蟺) 鈭(sinx/2 -cosx/2)^2 dx 鈭0<=x<=蟺 鈭0<=x/2<=蟺/2 褰0<=x/2<=蟺/4鏃 cosx/2>sinx/2 褰撓/4<=x/2<=蟺/2鏃 cosx/2<sinx/2 鈭 鍘熷紡=鈭(0...
绛旓細瑙ｆ硶璇疯涓嬪浘锛氬湪寰Н鍒涓紝鍑芥暟鐨涓嶅畾绉垎鏄竴涓〃杈惧紡锛屽畾绉垎鏄竴涓暟銆傦紝
绛旓細2銆佸洜涓簒^5+x^4-8=(x^3-x)(x^2+x+1)+(x^2+x-8)鎵浠(x^5+x^4-8)/(x^3-x)=(x^2+x+1)+(x^2+x-8)/x(x+1)(x-1)=(x^2+x+1)+(x^2+x)/x(x+1)(x-1)-8/x(x+1)(x-1)=(x^2+x+1)+1/(x-1)-[4/(x-1)-8/x+4/(x+1)]=(x^2+x+1)-3...
绛旓細绛旓細鈭 cot²x dx =鈭 cos²x / sin²x dx =鈭 (1-sin²x) /(sin²x) dx =鈭 (1/sin²x -1) dx =-cotx -x+C
绛旓細鐢ㄥ浘鐗囧ソ鐪嬶紝鍒嗗紑涓夐儴鍒嗗仛锛涓嶅畾绉垎 瀵兼暟锛氬畾绉垎+鏋侀檺锛
