一道微积分∫lnx/(1-x)²
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一道微积分
∫lnx/(1-x)²
∫lnx/(1-x)²
这种题目都不会?先凑微分再分部:
∫(lnx)/(1-x)² dx
= ∫(-lnx)/(1-x)² d(1-x)
= ∫lnxd[1/(1-x)]
= [1/(1-x)]lnx-∫[1/(1-x)]dlnx
= [1/(1-x)]lnx-∫[1/x(1-x)]dx
= [1/(1-x)]lnx+∫[1/(x-1)-1/x]dx
= [1/(1-x)]lnx+ln(x-1)-lnx+C
= ln(x-1)-[x/(x-1)]lnx+C
∫(ln(sinx))/sin²xdx
=-∫(lnsinx)dcotx
=-(lnsinx)cotx+∫cotxdlnsinx
=-(lnsinx)cotx+∫cot²xdx
=-(lnsinx)cotx+∫(csc²x-1)dx
=-(lnsinx)cotx-cotx-x+C
=-(lnsinx+1)cotx-x+C
欢迎追问、交流!
再问: 这3步没看懂,可以详细点吗.....
再答: ∫(lnsinx)/sin²xdx
= ∫(lnsinx)csc²xdx························ 余割运算定义cscx=1/sinx
=-∫(lnsinx)dcotx····························基本积分公式∫csc²xdx=-cotx+C
=-(lnsinx)cotx+∫cotxdlnsinx············ 分部积分公式∫udv=uv-∫vdu
=-(lnsinx)cotx+∫cotx(lnsinx)'dx
=-(lnsinx)cotx+∫cotx(1/sinx)(sinx)'dx
=-(lnsinx)cotx+∫cotx(1/sinx)cosxdx
=-(lnsinx)cotx+∫cotx(cosx/sinx)dx
=-(lnsinx)cotx+∫cot²xdx
欢迎继续追问!
∫(lnx)/(1-x)² dx
= ∫(-lnx)/(1-x)² d(1-x)
= ∫lnxd[1/(1-x)]
= [1/(1-x)]lnx-∫[1/(1-x)]dlnx
= [1/(1-x)]lnx-∫[1/x(1-x)]dx
= [1/(1-x)]lnx+∫[1/(x-1)-1/x]dx
= [1/(1-x)]lnx+ln(x-1)-lnx+C
= ln(x-1)-[x/(x-1)]lnx+C
∫(ln(sinx))/sin²xdx
=-∫(lnsinx)dcotx
=-(lnsinx)cotx+∫cotxdlnsinx
=-(lnsinx)cotx+∫cot²xdx
=-(lnsinx)cotx+∫(csc²x-1)dx
=-(lnsinx)cotx-cotx-x+C
=-(lnsinx+1)cotx-x+C
欢迎追问、交流!
再问: 这3步没看懂,可以详细点吗.....
再答: ∫(lnsinx)/sin²xdx
= ∫(lnsinx)csc²xdx························ 余割运算定义cscx=1/sinx
=-∫(lnsinx)dcotx····························基本积分公式∫csc²xdx=-cotx+C
=-(lnsinx)cotx+∫cotxdlnsinx············ 分部积分公式∫udv=uv-∫vdu
=-(lnsinx)cotx+∫cotx(lnsinx)'dx
=-(lnsinx)cotx+∫cotx(1/sinx)(sinx)'dx
=-(lnsinx)cotx+∫cotx(1/sinx)cosxdx
=-(lnsinx)cotx+∫cotx(cosx/sinx)dx
=-(lnsinx)cotx+∫cot²xdx
欢迎继续追问!
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