求解 ∫sin^3xcos^2xdx
作者&投稿:褚荀 (若有异议请与网页底部的电邮联系)
解:∫sin^3xcos^2xdx =-∫sin^2xcos^2xdcosx =-∫(1-cos^2x)*cos^2xdcosx =-∫(cos^2x-cos^4x)dcosx =(1/5)*cos^5x-(1/3)*cos^3x
如图
佐赖秘诀:[答案] ∫ (sin³x/cos²x)dx=-∫(sin²x/cos²x)d(cosx) = -∫(1-cos²x/cos²x)d(cosx) =-∫(1/cos²x - 1)d(cosx) = 1/cosx - cosx +C
章丘市19761169626: 微积分求解:∫sin^3 (x) cos^2 (x) dx 谢谢. - ?
佐赖秘诀: ∫sin^3 (x) cos^2 (x) dx =∫sin^2 (x) cos^2 (x)sin(x)dx =-∫sin^2 (x) cos^2 (x)dcos(x)=∫[cos^2 (x)-1]cos^2 (x)dcos(x) 设cos(x)=u 上式=∫(u^2-1)u^2du=∫(u^4-u^2)du=(u^5)/5-(u^3)/3+C=cos^5(x)/5-cos^3(x)/3+C
章丘市19761169626: 一道积分题 ∫sin^3xcos^3dx= - ?
佐赖秘诀: 如果sin^3x理解为(sinx)^3 ∫sin^3xcos^3dx=∫sin^3x(1-sin^2)dsinx =∫(sinx)^3dsinx-∫(sinx)^5dsinx =(sinx)^4/4-(sinx)^6/6+C
章丘市19761169626: ∫sin^3xcos^2xdx - ?
佐赖秘诀: ∫sin^2x cos^2x dx =∫1/4*(sin2x)^2 dx =∫1/4*(1-cos4x)/2 dx =1/8*∫(1-cos4x)dx =1/8*(x-1/4*sin4x+c) =x/8-sin4x/32+c
章丘市19761169626: ∫sin^3(x)cos^2(x)dx= - ?
佐赖秘诀: 把一个sin(x)拿出来 ∫sin^3(x)cos^2(x)dx=-∫sin^2(x)cos^2(x)d(cos(x))=-∫(1-cos^2)cos^2(x)d(cos(x))=-∫cos^2-cos^4(x)d(cos(x))=-(cos^3(x))/3+(cos^5(x))/5+c
章丘市19761169626: ∫(sinx^3/cosx^2)dx 不定积分 步骤清晰点 谢谢 - ?
佐赖秘诀: ∫(sinx^3/cosx^2)dx =∫(1-cosx^2)sinx/cosx^2dx =∫sinx/cosx^2-sinxdx=secx+cosx+c
章丘市19761169626: ∫sin^{2}xcos^{3}xdx= - ---- - ?
佐赖秘诀: ∫sin²xcos³xdx=∫sin²xcos²xd(sinx)=∫sin²x(1-sin²x)dsinx=∫sin²x-sin^4xdsinx=sin³x/3-sin^5x/5+C
章丘市19761169626: ∫(sinx+x^3)cos^2xdx - ?
佐赖秘诀: ∫sinx(cosx)^7a686964616fe78988e69d83313333376234632dx=-∫(cosx)^2dcosx=-(1/3)(cosx)^3 ∫x^3.(cosx)^2dx=(1/2)∫x^3.(1+cos2x)dx=(1/8)x^4 +(1/4)∫x^3.dsin2x=(1/8)x^4 +(1/4)x^3.sin2x -(3/4)∫x^2.sin2x dx=(1/8)x^4 +(1/4)x^3.sin2x +(3/8)∫x^2....
章丘市19761169626: 求e^3xcos2xdx不定积分,过程要详细点 - ?
佐赖秘诀: 这个用分步积分法 I=∫e^3xcos2xdx=1/3∫cos2xde^(3x)=1/3cos2xe^(3x)-1/3∫e^(3x)dcos2x=1/3cos2xe^(3x)+2/3∫e^(3x)sin2xdx=1/3cos2xe^(3x)+2/9∫sin2xde^(3x)=1/3cos2xe^(3x)+2/9sin2xe^(3x)-2/9∫e^(3x)dsin2x=1/3cos2xe^(3x)+2/9sin2xe^(3x)-4/9∫e^(3x)cos2xdx 整理得 I=∫e^3xcos2xdx=1/13[3cos2xe^(3x)+2sin2xe^(3x)]+C
章丘市19761169626: ∫x的3次方*cos(x的平方)dx 求详解 谢谢 - ?
佐赖秘诀: ∫x^3*cos(x^2)dx =(1/2)∫x^2cos(x^2)dx^2 =(1/2)∫x^2dsin(x^2) =(1/2)x^2sin(x^2)-(1/2)∫sin(x^2)dx^2 =(1/2)x^2sin(x^2)+(1/2)cos(x^2)+C